Biconditional is Self-Inverse

Theorem

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

where $\iff$ denotes the biconditional operator.

Proof
We apply the Method of Truth Tables.

As can be seen by inspection, the truth values under the main connective on the match those for $p$ on the  for all boolean interpretations:

$\begin{array}{|ccccc||c|} \hline (p & \iff & q) & \iff & q & p \\ \hline F & T & F & F & F & F \\ F & F & T & F & T & F \\ T & F & F & T & F & T \\ T & T & T & T & T & T \\ \hline \end{array}$