Biconditional is Commutative

Theorems
Equivalence is commutative:
 * $p \iff q \dashv \vdash q \iff p$

This can alternatively be rendered as:


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

These forms can be seen to be logically equivalent.

Proof by Natural Deduction
Commutativity is proved by the Tableau method:

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

Proof by Truth Table
We apply the Method of Truth Tables.

As can be seen by inspection, the truth values under the main connective matches for all models.

$\begin{array}{|ccc||ccc|} \hline p & \iff & q & q & \iff & p \\ \hline F & T & F & F & T & F \\ F & F & T & T & F & F \\ T & F & F & F & F & T \\ T & T & T & T & T & T \\ \hline \end{array}$