Biconditional is Self-Inverse

Theorem

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

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 LHS match those for $p$ on the RHS for all boolean interpretations:

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