$$(\exists x\in Z)(\forall y\in Z)(x>y)$$
This statement is true since we can take any $y\in Z$, add $1$ to it which would yield $x\in Z$ always greater than $y$.
If we now negate this statement we get: $$(\forall x\in Z)(\exists y\in Z)(x\le y)$$ This statement should be false, but if we take any $x\in Z$, add $1$ to it, we get $y\in Z$ such that $x\le y$ which makes the negation of a true statement a true statement??
Now there's probably something really wrong in my reasoning so can someone clarify this a little bit?