Biconditional is Commutative

Theorem

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

Proof

By the tableau method:

$p \iff q \vdash q \iff p$

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$p \iff q$	Proposition	(None)
2		1	$\left({p \implies q}\right) \land \left({q \implies p}\right)$	Material Equivalence	1
3		1	$\left({q \implies p}\right) \land \left({p \implies q}\right)$	Rule of Commutation	1
4		1	$q \iff p$	Material Equivalence	1

$\Box$

$q \iff p \vdash p \iff q$ is proved identically.

$\blacksquare$