Rule of Material Equivalence

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Theorem

The rule of material equivalence is a valid deduction sequent in propositional logic:

If we can conclude that $p$ implies $q$ and if we can also conclude that $q$ implies $p$, then we may infer that $p$ if and only if $q$.


Formulation 1

$p \iff q \dashv \vdash \paren {p \implies q} \land \paren {q \implies p}$

Formulation 2

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


Sources