In more detail: the underlying principle here is called De Morgan's law. De Morgan's law is our name for the fact that to say that a cat is not both furry and white, is the same as saying that the cat is either not-furry or not-white (or both).

(More generally: the negation of a conjunction (respectively, disjunction) is the disjunction (respectively, conjunction) of the negations.)

Suppose we lived in a world with twenty cats. We could make a statement about all of the cats by saying "The first cat is furry and the second cat is furry and the third cat is furry and [...] and the twentieth cat is furry." But that would take too long; instead we just say, "Every cat is furry." Similarly, instead of "Either the first cat is white or the second cat is white or [...] or the twentieth cat is white," we can say, "There exists a white cat." Thus, the same principles that we use for and-statements ("conjunctions") and or-statements ("disjunctions") can be used on ("quantified") for every-statements and there exists-statements. "There does not exist a winged cat" is the same thing as "For every cat, that cat does not have wings" for the same reason that "It is not the case that either the first cat has wings or the second cat has wings" is the same thing as "The first cat does not have wings and the second cat does not have wings." That's de Morgan's law.

So, suppose there does not exist a person who does not die. De Morgan's law tells us that this is equivalent to saying that for every person, that person does not-not-die. But not-not-dying is the same thing as dying. But this is that which was to be proven.