Actually all names were written in the Book of Life since God offers eternal life to all. However those who choose to disobey the Gospel, reject the Gospel, and reject Christ are EXPUNGED (blotted out) of the Book of Life.
And the LORD said unto Moses, Whosoever hath sinned against me, him will I blot out of my Book. (Exodus 32:33)
So if you were thinking this supports unconditional election, it does not. Unconditional election for salvation is NOT taught in Scripture. It is a purely man-made doctrine.
So now the question is "Why do people love man-made doctrines over Gospel truth?
Not according to scripture:
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"And all who dwell on the earth will worship the beast—
all whose names have not been written from the foundation of the world in the Book of Life belonging to the Lamb who was slain."
And those who dwell on the earth
whose names were not written in the Book of Life from the foundation of the world will marvel when they see the beast that was, and is not, and yet will be.
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Simply put, God wrote all of the names of those He had chosen in the book of life before the world even began. Likewise, there were names that were not written in the book of life before the world began.
There is no scriptures that teach that every persons name was written in the book of life and then blotted out. It would go against what the scripture above is stating, which is that their names were never even written in the book of life.
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"Like them, he who overcomes will be dressed in white. And I will never blot out his name from the Book of Life, but I will confess his name before My Father and His angels."
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If we reverse the scripture above, those who do not overcome with their garments remaining soiled, their names can and will be blotted out of the book of life. As believers in Christ, even though we have been chosen for salvation, we still need to have faith. We are saved by grace through faith. Therefore, faith is our part. If we soil our garments by going back into the world and willfully live according to the sinful nature and die in that state, then the Lord can and will blot out those names.
But getting back to the issue, only the names of those whom God chose from before the world began were written in the book of life. Those whom He did not chose, their names were never written in there.
Salvation is by God's Sovereign choice according to His will and not by man's efforts or desires, as Paul made clear. This truth is not man-made, but is according to scripture.