Properties of Biconditional

Theorem

Rule of Material Equivalence

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

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

Biconditional as Disjunction of Conjunctions

Formulation 1

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

Formulation 2

$\vdash \paren {p \iff q} \iff \paren {\paren {p \land q} \lor \paren {\neg p \land \neg q} }$

Biconditional Equivalent to Biconditional of Negations

Formulation 1

$p \iff q \dashv \vdash \neg p \iff \neg q$

Formulation 2

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

Biconditional iff Disjunction implies Conjunction

Formulation 1

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

Formulation 2

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