How do Trinitarians explain John 17:3?

First of all, Jesus addressing the Father as "You, the only true God" does not semantically mean "that no one else is true God except the Father." That could be inferred only if Jesus had said "You, Who alone are the only true God". It is most evident that the statement "the Father is the only true God" does not imply the statement "the Father alone is the only true God".

Addressing the question, the meaning of Jesus calling the Father "the only true God" is just to confirm that there is only one omnipotent eternal God, Creator, Sustainer and Lord of everything that exists outside Him, which is a truth accessible by the natural light of human reason (Rom 1:19-21) and explicitely revealed in the OT:

"You are my witnesses," declares the LORD, "and my servant whom I have chosen, so that you may know and believe Me and understand that I am He. Before Me there was no God [el] formed, neither shall there be after Me. (Is 43:10)

"Thus says YHWH, the King of Israel and his Redeemer, YHWH of hosts: 'I am the first and I am the last, and there is no God besides Me. (Is 44:6)

Do not tremble and do not be afraid; have I not told you from of old and declared it? And you are my witnesses! Is there a God [eloah] besides Me? There is no Rock; I know not any.'" (Is 44:8)

"I am YHWH, and there is no other; besides Me there is no God. I will gird you, though you have not known Me; (Is 45:5)

"Declare and set forth your case; indeed, let them consult together. Who has announced this from of old? Who has long since declared it? Is it not I, YHWH? And there is no other God besides Me, a righteous God [el] and a savior; there is none except Me. (Is 45:21)

Now, it is well-known that the name Elohim, literally "the gods", has two meanings in the OT: a) the only omnipotent, eternal Creator God, YHWH, in which case the plural has a majestatic sense and the name is the subject of a singular verb, and b) the gods, either existing super-human entities created by and subordinated to YHWH (in pre-exilic texts only, of which I make the case below) or the imaginary gods of the gentiles, in which case is the subject of a plural verb.

Thus, verses like Is 44:6 and 45:5 which say literally "besides Me no Elohim/elohim" can be understood in either of two senses, depending on the sense of Elohim/elohim: with "Elohim" in sense a, as "besides me no omnipotent, eternal Creator God", and with "elohim" in sense b, as "besides me no gods", so that someone intent on affirming that post-exilic Jews did NOT reserve the term "god" for YWHW can reject these verses as proof to the contrary by saying that they use Elohim in sense a only, and therefore do not preclude the existence of lesser, subordinated, created gods which should not be worshipped.

Therefore I emphasized above the instances where "God" translated a singular Hebrew name, either "el" (Is 43:10) or "eloah" (Is 44:8), which I gave between []. From these instances, it is clear that for post-exilic Jews all names referring to a divinity, both the majestatic plural "Elohim", which in Greek would be "ho Theos", and the singular "El" and "Eloah", which in Greek would be "Theos", were reserved to YHWH. The other existing superhuman real entities, the angels, were created by YHWH and wholly subordinated to Him, so that they could not be called "gods". And the "gods" of the idolatrous peoples were not real: "For all the gods [elohe] of the peoples are idols, but YHWH made the heavens" (Ps 96:5).

I wrote the 3 previous paragraphs to preemptively dispel the notion that in Jn 1:1 the final unarthrous "Theos" is actually "theos" and refers to a created superhuman entity, as non-trinitarians (or more precisely, as people denying the numerical identity of ousia between the Father and the Son) posit. But Jesus calling the Father "the only true God" in Jn 17:3 achieves exactly the same: because if the Father is "the only true God", then there are only two possible ways to understand the final clause "the Word was God" (or "the Word was god") in Jn 1:1:

A. The h*m*ousian way: the Word is all that God the Father is (except Father), i.e. the Word is also the only true God;

B. The heteroousian way: the Word is a fake god.

So, while the meaning of Jesus calling the Father "the only true God" is quite straightforward and not new to the NT, the reason why John recorded it is to preemptively dispel any heteroousian reading of the passages where Jesus is called theos. Because if there is only one true God, the Father (also in 1 Cor 8:6), and the Son is not all that the Father is (except Father), then the Son is a fake God.

Since I have used the noun ousia by itself and composed with h*m*/hetero, I note that it is the noun directly deriving from the verb "to be" in Greek, first person singular "eimi" and present participle "on", meaning what a subject is, and that Jesus used that verb to state his divinity in John's Gospel in two ways: 4 times with Himself as subject without further qualifications, as "I Am", "Ego Eimi", clearly in the sense of the Name of God in the first person revealed in Ex 3:14, in Jn 8:24,28,58 and Jn 13:19, and once with Himself and the Father as subject, "I and the Father are one" in Jn 10:30. Refraining from using a noun which derives directly and naturally from a key verb used in the Gospel is unnatural and unwarranted.

To place the above in the context of other NT passages containing "Theos", we must first note that the NT uses two terms, "ho Theos" and unarthrous "Theos".

The term "ho Theos" or its genitive "tou Theou", dative "to Theo", or accusative "ton Theon", refer to God the Father, except in the 5 passages where it refers to the Son, none of which calls Jesus simply "ho Theos" without qualification: Mt 1:23, Jn 20:28, Ti 2:13, 2 Pe 1:1, 1 Jn 5:20. But even in these cases "ho Theos" refers always to a divine Person, not to the divine ousia.

On the other hand, unarthrous "Theos" can refer to either

• the one and only divine ousia, what each divine Person Is, in which case it is the attribute of a copulative sentence whose subject is the Son (Jn 1:1, Rom 9:5) or the subject of a passive predicative sentence, or

• a divine Person, usually God the Father when it appears without qualification or the Son in "monogenēs Theos" (Jn 1:18).

So, Jn 1:1 says:

"In the beginning was the Word, and the Word was with God the Father, and the Word was all that God the Father was (except Father)."

Where from monotheism, "all that God the Father was" is understood in a sense of numerical identity, not of merely qualitative identity. "I and the Father are one" (Jn 10:30), not "I and the Father are equal". "Homoousios", i.e. "of the same ousia" (numerical identity), not "isoousios", i.e. "of identical ousiai" (merely qualitative identity).

In terms of the four possible theological positions: consubstantial Trinity (nicene orthodoxy), tritheism, Arianism and modalism:

• Modalism is directly discarded by holding the Gospel text as true.

• Tritheism is discarded by the OT and also by Jn 17:3 (and 1 Cor 8:6): there are not two or more true Gods.

Thus, you can either hold the consubstantial Trinity, whereby the Son is also the only true God, the Father's design "that all may honor the Son, even as they honor the Father" (Jn 5:23) implies worshipping the only true God, and the statement "I and the Father are one" is meant in an ontic sense,

or hold Arianism, whereby the Son is a fake god, the Father's design "that all may honor the Son, even as they honor the Father" (Jn 5:23) implies divinely mandated idolatry, and the statement "I and the Father are one" is meant in a merely moral sense.

Monotheism
A belief in the Trinity is a belief in monotheism, which "is defined as the belief in the existence of only one god that created the world, is all-powerful and intervenes in the world."¹ The belief in a single God, absolute monotheism, who is responsible for creating all things is distinctly Jewish:

The doctrine of absolute monotheism is preached in the most emphatic manner by Jeremiah (x. 10; xiv. 22; xxiii. 36; xxxii. 18, 27) and the Deuteronomist(iv. 35, 39), but the Biblical teaching on the subject may be said to have culminated in Isaiah of Babylon. Yhwh, though in a peculiar sense the God of Israel, is still the God of all the world. This prophet's standpoint is uncompromising: "I, even I, am the Lord; and beside me there is no savior" (xliii. 11); "I am the first, and I am the last; and besides me there is no God" (xliv. 6, xlviii. 12); "that they may know from the rising of the sun to the setting thereof that there is none besides me; I am God and there is none else" (xlv. 6, Hebr.)²

Jewish monotheism is "encapsulated in the first verse of the Shema: "Hear, O Israel: the LORD our God, the LORD is one" ³

“Hear, O Israel: The LORD our God, the LORD is one." ^[a] (Deuteronomy 6:4) [ESV]
_{[a] Deuteronomy 6:4 Or The LORD our God is one LORD; or The LORD is our God, the LORD is one; or The LORD is our God, the LORD alone.}

שְׁמַע יִשְׂרָאֵל יְהוָה אֱלֹהֵינוּ יְהוָה אֶחָֽד

As shown in the ESV note, it is possible to understand "the LORD is one" as "the LORD alone." In that case John 17:3 could be construed to have some affinity with the Shema:

And this is eternal life, that they know you, the only true God, and Jesus Christ whom you have sent. (John 17:3)

"Only" is μόνος which, in the King James, is translated as "only" 24 times and as "alone" or "by one's self" 23 times. In other words, there is an equal number of times when the word conveys "only" in the sense of physical separation causing one to be "by one's self" that is, to be "alone."

For example immediately before His prayer Jesus used μόνος where it is understood as "alone."

Behold, the hour is coming, indeed it has come, when you will be scattered, each to his own home, and will leave me alone. Yet I am not alone, for the Father is with me. (John 16:32)

However, that is not how the Greek language is used to translate the passage in Deuteronomy which is used in the traditional expression of Jewish monotheism:

And these are the statutes and judgments which the Lord commanded to the sons of Israel in the wilderness as they were coming out from the land of Egypt. Hear O Israel: The Lord our God is one Lord." (Deuteronomy 6:4) LXX-NETS

καὶ ταῦτα τὰ δικαιώματα καὶ τὰ κρίματα ὅσα ἐνετείλατο κύριος τοῗς υἱοῗς Ισραηλ ἐν τῇ ἐρήμῳ ἐξελθόντων αὐτῶν ἐκ γῆς Αἰγύπτου ἄκουε Ισραηλ κύριος ὁ θεὸς ἡμῶν κύριος εἷς ἐστιν

The word used for "one" is εἷς not μόνος. Jesus affirmed the first verse of the Shema as the most important commandment and He used the same language as the LXX:

And one of the scribes came up and heard them disputing with one another, and seeing that he answered them well, asked him, “Which commandment is the most important of all?” Jesus answered, “The most important is, ‘Hear, O Israel: The Lord our God, the Lord is one.
(Mark 12:28-29)

καὶ προσελθὼν εἷς τῶν γραμματέων ἀκούσας αὐτῶν συζητούντων ἰδὼν ὅτι καλῶς ἀπεκρίθη αὐτοῖς ἐπηρώτησεν αὐτόν ποία ἐστὶν ἐντολὴ πρώτη πάντων ἀπεκρίθη ὁ Ἰησοῦς ὅτι πρώτη ἐστίν ἄκουε Ἰσραήλ κύριος ὁ θεὸς ἡμῶν κύριος εἷς ἐστιν

The proper expression of an absolute God would be "the one true God" or simply "the one God." If Jesus desired to emphasize the singular nature of the true God the traditional word to use would be εἷς (one) not μόνος. Significantly, the word Jesus added was one whose meaning is ambiguous.

Therefore, John 17:3 cannot be approached as a traditional expression of Jewish monotheism. Rather, Jesus made a new profession of His belief in absolute monotheism.

The Alone True God - τὸν μόνον ἀληθινὸν θεὸν
Since there is one God, then the true God is God, and it is unnecessary to embellish the phrase:

And we know that the Son of God has come and has given us understanding, so that we may know him who is true; and we are in him who is true, in his Son Jesus Christ. He is the true God and eternal life. (1 John 5:20)

Arguably, "He" in "He is the true God and eternal life" refers to Jesus Christ, but those who deny the Trinity would say it refers to "His" in "His Son Jesus Christ." However, the key is John omits μόνος from the phrase. The "only true God" in the Gospel has become simply the "true God" in the Letter:

• In the Gospel Jesus included μόνος because it was necessary at the time
• In the letter John removed μόνος because it was no longer necessary

Jesus' new expression of monotheism is evidence of the Trinity:

And this is eternal life, that they know you [My Father] the [temporarily] by one's self/alone true God, and Jesus Christ whom you have sent.

At the time Jesus was praying μόνος is necessary because God was alone: the Father had sent the Son who was away. When John writes the letter, they are one and μόνος is no longer appropriate because John believes Jesus and the Father are once again together (cf. John 1:18).

Notes:
1. Monotheism
2. Jewish Monotheism
3. Shema

A word of caution: You explicitly asked for a Patristic source and the Greek east does not see eye to eye with the Latin west on the Trinity. For example the East rejects the filioque from the 2nd Nicene council,and they did not like the use of substance (ousia) and preferred to talk about the members of the Trinity as Person (hypostasis). These suggest you may get a different answer if you ask someone from the Orthodox church.

That being said Augustine (who wrote during the Patristic era, but in Latin) explicitly talks about this passage in several places in De Trinitate. Here is an excerpt from Book VI Chapter 2 sections 10 and 11 (VI.2.10-11) (Augustine is verbose so I won't quote the entire thing)

Now that we have shown how it is possible to talk about the "Father alone" in the sense that none none but he is the Father, we must go on to examine the opinion that the only true God (Jn 17:3) is not the Father alone, but Father and Son and holy Spirit...He was speaking to the Father and he had named the Father he was speaking to when he said This eternal life, that they should know you the one true God (Jn 17:3) the Arians like to take this meaning that the Son is not true God...

He eventually formulates an argument against the Arians (in the same section)

But here again if only the three all together can be called God, how is God the head of Christ--that is, on this supposition, the trinity the head of Christ--when Christ is included in the trinity to make it three? Or is it t that what the Father and Son are together is head of what the Son is alone? The Father and Son together are God, but only the Son is Christ, especially as it is the Word already made flesh who is speaking in the lowliness by which the Father is greater, as he says himself, For the Father is greater than I (Jn 14:28). So it might be that his being God, which he has in common with the Father, is head of the man mediator which he alone is....the Word which is God together with the Father is the head of Christ, although the man can only be understood as Christ together with the Word which became flesh?" (Edmund Hill translation)

I hope this helps.

That scripture verse that you mention was used by Arius to maintain that Jesus was not of same nature as Father. However, his objection was resolved.

St. Thomas explains the difficulty you mention in his commentary on the gospel of John 17; on this point he writes:

Secondly, we should explain the phrase, you the only true God. It is clear that Christ was speaking to the Father, so when he says, you the only true God, it seems that only God the Father is true God. The Arians agree with this, for they say that the Son differs by essence from the Father, since the Son is a created substance, although he shares in the divinity more perfectly and to a greater degree than do all other creatures. So much more that the Son is called God, but not the true God, because he is not God by nature, which only the Father is.

Hilary answers this by saying that when we want to know whether a certain thing is true, we can determine it from two things: its nature and its power. For true gold is that which has the species of true gold; and we determine this if it acts like true gold. Therefore, if we maintain that the Son has the true nature of God, because the Son exercises the true activities of divinity, it is clear that the Son is true God. Now the Son does perform true works of divinity, for we read, "Whatever he [the Father] does, that the Son does likewise" (5:19); and again he said, "For as the Father has life in himself," which is not a participated life, "so he has granted the Son also to have life in himself" (5:26); "That we may be in his true Son, Jesus Christ. This is the true God and eternal life" [1 Jn 5:20].

According to Hilary, he says, you the only true God, in a way that does not exclude another. He does not say without qualification, you the only, but adds and Jesus Christ whom you have sent. It is like saying: that they know you and Jesus Christ whom you have sent to be the one and only true God. This is a pattern of speaking that we also use when we say [in the Gloria]: "You alone, Jesus Christ, are the most high, together with the Holy Spirit." No mention is made of the Holy Spirit because whenever the Father and the Son are mentioned, and especially in matters pertaining to the grandeur of the divinity, the Holy Spirit, who is the bond of the Father and Son, is implied.

Or, according to Augustine in his work, The Trinity, he says this to exclude the error of those who claim that it is false to say that the Father is God, and the Son is God, and the Holy Spirit is God; while it is true to say that the Father and the Son and Holy Spirit are one God. The reason for this opinion was that the Apostle said that "Christ [is] the power of God and the Wisdom of God" (1 Cor 1:24). Now it is clear that we cannot call anyone God unless he has divine power and wisdom. Therefore, since these people held that the Father was wisdom, which is the Son, they held further that the Father considered without the Son would not be God. And the same applies to the Son and the Holy Spirit.

Also in Summa Contra Gentiles St. Thomas Aquians sets up various objections (that Arius used) against opinion that Jesus is God here (among objections is one that uses John 17:3.) He resolves them here. It is benifical to read objections and answers for detailed solutions to the objections.

St. Thomas argues that in God there is begetting, fatherhood and sonship in book IV of Summa contra gentiles (see chapters from 1 to 14) where he also refutes opinions of Protinus and Sabellius.

The original question contains an implied logical fallacy: If God the Father is the One true God then He is the only God. I will now show that this statement is incorrect according to the Bible.

1. The Bible in numerous places vehemently teaches the doctrine of monotheism, for example: Deut 4:35, 6:4, 32:39, Isa 44:6, 45:5, 6. It is also taught in the NT just as strongly: 1 Cor 8:4; Eph 4:6, 1 Tim 1:17, John 17:3. Let there be no doubt that there is one and one only God, according to the Bible. For example, Isa 45:5, "I am the LORD, and there is no other; apart from me there is no God." Despite the above, the NT repeatedly calls Jesus God, Matt 1:22, 23; John 1:1, 18, Col 2:9, and not just any god, but uses the phrase, "ho theos" so that there can be no doubt, Matt 20:28, Heb 1:8. The implication here is that both God the Father and Jesus are the one true God.
2. We have the same situation with the title, "Creator". For example in Isa 44:24, "I am the LORD, the Maker of all things, who stretches out the heavens, who spreads out the earth by myself" See also Isa 45:18. Yet in the NT we have similar assertions that Jesus created all things, John 1:3, Col 1:16, 17. Again, the implication is that both Jesus and God constitute the One true creator.
3. There is an almost identical situation with Saviour and Redeemer; for example, Isa 43:11 says: "I, even I, am the LORD, and apart from me there is no savior." See also Isa 43:3, 45:18. Yet in the NT we often have Jesus referred to our One true Saviour and Redeemer: Matt 1:21; Acts 4:12; 2 Tim 1:10; Tit 1:4, 2:13, 3:6; 2 Pet 1:1, 11. Once more, the implication being that Both God the Father (YHWH) and Jesus constitute the One true saviour of mankind.
4. The same is true of the One Rock as per Isa 44:8, "Is there any God besides me? No, there is no other Rock; I know not one." See also Deut 32:3,4,15; Ps 92:15. In the NT we find that 1Cor 10:4 says, "for they drank from the spiritual rock that accompanied them, and that rock was Christ." The implication is that both God the Father and Jesus constitute the One true Rock.
5. And so we could go one. For example God the Father does not share His glory (Isa 42:8, 48:11) yet we find in John 17:5, 24 Jesus shares the Father's glory.
6. In Isa 41:4, 44:6 The Father (YHWH, LORD) is called, "First and Last" and so is Jesus, Rev 1:17, 18, 22:13
7. Jehovah is the declared the only One who is deserving of worship in Ex 20:3, 34:14; Deut 8:19; 2 Kings 17:35-38; (see also Matt 4:10; Acts 10:25, 26; Rev 19:10, 22:8, 9), yet the NT records numerous times when Jesus was worshipped Matt 2:11, 14:33, 28:9, 17; Luke 4:8; 24:52; John 9:38; Rom 10:9, Heb 1:5, 6, Phil 2:10, etc.

The above is only a sample of many more. In all these cases, the fact that the Bible calls God the Father the One true God does not prevent Jesus being the One true God. Note that this does not make Jesus ALSO God - definitely not. It simply means that both (at least) Jesus and the Father are the One true God. A similar analysis would show that the same can be said of the Holy Spirit. This is the central assertion of the doctrine of the Trinity - One God, three persons. Therefore, John 17:3 is absolutely consistent with this teaching.

However, by itself John 17:3 is a proof text for neither Arians nor Trinitarians.