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When God warned Adam about eating from the tree of the knowledge of good and evil he said that in the day that he ate there of he would surely die.
Genesis 2:17 KJV
But of the tree of the knowledge of good and evil, thou shalt not eat of it: for in the day that thou eatest thereof thou shalt surely die.
We know that Adam lived over 900 years on the Earth:
Genesis 5:5 KJV
And all the days that Adam lived were nine hundred and thirty years: and he died.
So Adam and Eve did not die physically that day, but the argument could be made that they did die in the respect that they were now separated from God for eternity.
Since God had created man in their image, which I take to mean as in three parts body, spirit, and soul.
If sin had separated man from God eternally, and man was condemned to eternity in the lake of fire that would be the second death:
Revelation 21:8 KJV
But the fearful, and unbelieving, and the abominable, and murderers, and whoremongers, and sorcerers, and idolaters, and all liars, shall have their part in the lake which burneth with fire and brimstone: which is the second death.
And since man was condemned to the lake of fire for eternity, which is the second death that would have taken place on the same day as their sin.
Could that have been the reason why:
Genesis 3:22 KJV
And the LORD God said, Behold, the man is become as one of us, to know good and evil: and now, lest he put forth his hand, and take also of the tree of life, and eat, and live for ever: