It's not eternal life as an eternal life with Christ, but that of an eternal torment in the sense of spending an eternal miserable life being tormented day and night. No, scripture never calls it an eternal life in the lake of fire, but the scenario would be that of the complete opposite of the eternal life with Christ. Existing is still life, whether it is a glorified life in Christ (New Jerusalem) or one of damnation and torment in the lake of fire as both are eternal.
One needs to define a word to understand what it means. Scripture never says what we will look like in the resurrection of life or damnation. Matthew 13:42 says there will be weeping and gnashing of teeth from those who are cast into the lake of fire so that one verse alone shows me they will have life, but no one knows what they will look like. The rest of these verses are pertinent to that of the resurrection of life both in the New Jerusalem and the lake of fire.
Exist: to have actual being or reality; to live
Existence: the fact or state of existing, living or occurring the manner of existing
Genesis 2:7 And the LORD God formed man of the dust of the ground, and breathed into his nostrils the breath of life; and man became a living soul.
Ecclesiastes 12:7 Then shall the dust return to the earth as it was: and the spirit shall return unto God who gave it.
Dust + breath from God = a living soul. When the flesh dies physically it returns back to the dust of the ground and our spirit is preserved with God for final judgment.
John 5:28 Marvel not at this: for the hour is coming, in the which all that are in the graves shall hear his voice,
John 5:29 And shall come forth; they that have done good, unto the resurrection of life; and they that have done evil, unto the resurrection of damnation.
1Co 15:50 Now this I say, brethren, that flesh and blood cannot inherit the kingdom of God; neither doth corruption inherit incorruption.
1Co 15:51 Behold, I shew you a mystery; We shall not all sleep, but we shall all be changed,
1Co 15:52 In a moment, in the twinkling of an eye, at the last trump: for the trumpet shall sound, and the dead shall be raised incorruptible, and we shall be changed.
Mat 13:40 As therefore the tares are gathered and burned in the fire; so shall it be in the end of this world.
Mat 13:41 The Son of man shall send forth his angels, and they shall gather out of his kingdom all things that offend, and them which do iniquity;
Mat 13:42 And shall cast them into a furnace of fire: there shall be wailing and gnashing of teeth.
Act 24:15 And have hope toward God, which they themselves also allow, that there shall be a resurrection of the dead, both of the just and unjust.
Rev 20:12 And I saw the dead, small and great, stand before God; and the books were opened: and another book was opened, which is the book of life: and the dead were judged out of those things which were written in the books, according to their works.
Rev 20:13 And the sea gave up the dead which were in it; and death and hell delivered up the dead which were in them: and they were judged every man according to their works.
Rev 20:14 And death and hell were cast into the lake of fire. This is the second death.
Rev 20:15 And whosoever was not found written in the book of life was cast into the lake of fire.
One can not compare eternal life with the Father with eternal damnation in the lake of fire except that the two are opposite from each other. Those who are resurrected to be with the Lord will have an eternal life (existence) basking in Gods glory. Those who are cast in the lake of fire will not literally be consumed by an eternal literal fire as fire here means torment, destruction or as being destroyed or broken down like someone saying that we are going to destroy you and drag your name through the mud.
They are not literally destroyed, but brought down to a lower level that separates them from God forever. Those in the lake of fire will have an existence of weeping and gnashing of teeth for the destruction and torment they will face for eternity. Death means separation from God as their eternal destiny is the lake of fire. If death here meant not to exist anymore than what would be the purpose of the second death and being resurrected to stand in the great white throne judgment. Couldn't God just make them vanish as if they never existed in the first place!
Death is an existence and if one is existing then it has to be some form of life. Not like the life we have now as being alive here on earth, not like eternal life with the Father, but yet a form of a tormented eternal life of existence in the lake of fire for eternity.