That's correct, which proves it is possible to
fall away from the faith back unto
death by ongoing sin and have your name blotted out of the book of Life. Hyper grace and OSAS followers will try to twist the meaning of this verse, as they often do with a lot of other scriptures in order to make it fit into their false doctrine. They will claim everyone who was ever been born has his name in the book of life, which is totally
false. Only those who have passed from death
to Life are written in the book of Life, as all men were once "dead" in their sins and trespasses.
Those who have their names blotted out of the book of Life are known in scripture as being
twice dead. The twisting and perverting of Gods words by teaching lies can also get one blotted out of the book of Life, not to mention ongoing sin.
Revelation 21:27
And there shall in no wise enter into it any thing that defileth, neither whatsoever
worketh abomination, or
maketh a lie: but they which are written in the Lamb's book of life.
Revelation 22:19
And if any man shall
take away from the words of the book of this prophecy, God shall
take away his part out of the book of life, and out of the holy city, and from the things which are written in this book."