De Morgan's Laws (Predicate Logic)/Denial of Universality/Formulation 2

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

Then:
 * $\vdash \neg \paren {\forall x: \map P x} \iff \paren{ \exists x: \neg \map P x }$

This can be expressed as two separate theorems: