Let's assume that I am Abraham Lincoln. Mr. Lincoln is dead. Dead people cannot write. Therefore, this blog was never written.
Since this blog is written and you are reading it, I am clearly not Abraham Lincoln. (For our non-American readers, Mr. Lincoln was America's 16th president, who died in 1865.)
My favorite use of this was Bertrand Russel's argument about how if one allowed enough false assumptions, one could prove any absurd thing. His student asked him to prove that he was the pope (laughable, as Mr. Russel was an atheist) given that zero and one were the same number. (This assumption would have many other hilarious implications.)
Since 0 equals one, I can add one to both sides to get the equation 2=1. The pope and I are two separate individuals. But since 2=1, I and the pope are actually the same individual. Therefore, I am the pope.