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

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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:

Forward Implication

$\vdash \neg \paren {\forall x: \map P x} \implies \paren{ \exists x: \neg \map P x }$


Reverse Implication

$\vdash \paren{ \exists x: \neg \map P x } \implies \neg \paren {\forall x: \map P x}$


Proof


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