Exclusive Or with Itself

Theorem
Exclusive or destroys copies of itself:
 * $p \oplus p \dashv \vdash \bot$

Proof
We apply the Method of Truth Tables.

As can be seen by inspection, the truth values under the main connective match for each boolean interpretations.

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