Non-Equivalence as Equivalence with Negation

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Theorem

Formulation 1

$\neg \paren {p \iff q} \dashv \vdash \paren {p \iff \neg q}$

Formulation 2

$\vdash \neg \left ({p \iff q}\right) \iff \left({p \iff \neg q}\right)$