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

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

Then:
 * $\neg \paren {\forall x: \map P x} \dashv \vdash \exists x: \neg \map P x$
 * If not everything is, there exists something that is not.