If I have $\forall x: P(x)$, what is the equivalent function if I attach a not to the whole function:
$\neg (\forall x: P(x))$
Read the statement out: not (for all x, p(x) holds). So if not for all x, there must be some x such that p(x) doesn't hold. So:
Then you have $\exists x : \lnot P(x)$.