Jesus is eternal life, He is life. We see this over and over again in the Apostle Johns writings. In Him was LIFE and that LIFE was the light of man. Life of the world, the Bread of LIFE,My words are spirit and they are LIFE, I AM the way,the truth and the LIFE,The LIFE was manifested and we proclaim to you the ETERNAL Life which was with the Father, and was manifested to us- The Prologue of 1st John. So we see that when John uses the phrase True God and Eternal Life together in 1 John 5:20 that He is referring to Christ as the closest antecedent making Him the True God and Eternal Life.
Also we see that when we search the GNT that Eternal Life is never used of the Father without the Son but we see that Eternal Life is used over and over with Jesus where the Father is never mentioned. This makes a solid case for Jesus in 1 John 5:20 as the True God and Eternal Life.
1 John 5:20
20 We know also 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 — even in his Son Jesus Christ. He(Jesus) is the true God and eternal life.
On behalf of seeing χριστος as the antecedent are the following arguments: (1) Although it is true that αληθινος θεος is not elsewhere referred to Christ, αληθεια is, and is so in Johannine literature (John 14:6).
29 Winer-Moulton, 195.
Further, αληθινος θεος is not a "constant.. epithet" as Winer supposes, being found only in John 17:3 and 1 John 5:20! (2) Christ is also said to be ζωη in John's writings John 11:25; 14:6; 1 John 1:1-2), an epithet nowhere else used of the Father. (3) The demonstrative pronoun, ουτος, in the Gospel and Epistles of John seems to be used in a theologically rich manner.30 Specifically, of the approximately seventy instances in which ουτος has a personal referent, as many as forty- four of them (almost two-thirds of the instances) refer to the Son. Of the remainder, most imply some sort of positive connection with the Son.31 What is most significant is that never is the Father the referent.For what it is worth, this datum increases the probability that ιησου χριστω is the antecedent in 1 John 5:20. 32 The issue cannot be decided on grammar alone. But suffice it to say here that there are no grammatical reasons for denying that αληθινος θεος is descriptive of Jesus Christ.
1st- Jesus is called God in the writings of John(1:1,20:28,1 John 5:20)
2nd- Jesus is called Eternal Life over and over again in Johns writings
3rd- John opens up his epistle with the Eternal life(Jesus) that was with the Father in the beginning and was manifest to the disciples(1 John 1:1-5)
4th- John ends his epistle with Jesus who is eternal life and only is eternal life found in Him who is the true God.
5th- never is eternal life used of the Father alone. When the Father is included the Son is always mentioned together with the Father making them equal. Equality with the Father was not something Jesus needed to grasp at as He already possessed complete Deity as God.(Phil 2, Col 2:9)
6th- John would not leave his readers with any ambiguity warning them to guard themselves from idols(5:21) So this would be clear his reference was to those who reject Jesus as the true God. They are the idoloters and antichrists John writes of in his epistles.
7th- Jesus is also the True God and the True one in 1 John. Jesus is the true light which brings light to all men (John 1:9) Jesus is the truth (John 14:6)Jesus is the true vine (John 15:1). Jesus is the true witness of God (John 18:37) He who is true (Revelation 3:7) Jesus is the faithful and true witness (Revelation 3:14)Jesus is Lord God Almighty, Just and true are your ways(Revelation 15:3) Jesus is faithful and true(Revelation 19:11).
8th- [In John's writings] Of the approximately 70 instances in which ουτος has a personal referent, as many as 44 of them (almost 2/3 . . . ) refer to the Son. Of the remainder, most imply some sort of positive connection with the Son.31 What is most significant is that NEVER is the Father the referent.FWIW, this datum increases the probability that ιησου χριστω IS the antecedent in 1 John 5:20. Wallace.
9th- Guard yourself from idols. Christ is God and not an idol so its ok to worship Him as the True God and Eternal life.
10th- The disciples all worshiped Christ as God and called Him God in their writings which agrees with 1 John 5:20
So the most logical conclusion is that it refers to Jesus as the true God. Not only is this Wallace's conclusion from Johns usage of outos but He is the closest antecedent (most times in the GNT this principal holds true). Eternal Life is never used of the Father alone in John’s writings and only a couple of times does John include the Father with the Son regarding eternal life. John opens up his epistles describing the "eternal life" who was with the Father in the beginning and then ends his epistle with eternal life identifying Jesus as the true God and eternal life. John then says this in the last verse:
1 John 5:21-Guard yourselves from idols
Now why would he leave any ambiguity in verse 20 as to the identity of the true God and eternal life then turn around commanding them to protect themselves from idolatry ? It make absolutely no sense whatsoever unless John is identifying Jesus as the true God and refuting the Gnostics of his day who denied the Incarnation.
hope this helps !!!