Exclusive Or with Contradiction/Proof 2

Theorem

 * $p \oplus \bot \dashv \vdash p$

Proof
We apply the Method of Truth Tables to the proposition.

As can be seen by inspection, in each case, the truth values in the appropriate columns match for all boolean interpretations.

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