De Morgan's Laws (Predicate Logic)/Denial of Universality

Theorem
Let $\forall$ and $\exists$ denote the universal quantifier and existential quantifier respectively.

Then:
 * $\neg \forall x: P \paren x \dashv \vdash \exists x: \neg P \paren x$


 * If not everything is, there exists something that is not.

Proof
So $\exists x. \neg P \paren x \vdash \neg \forall x.P \paren x$