Exclusive Or is Self-Inverse

Theorem

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

where $\oplus$ denotes the exclusive or 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:

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