# Biconditional is Reflexive

## Theorem

The biconditional operator, considered as a relation, is reflexive:

$\vdash p \iff p$

This can otherwise be stated as that equivalence destroys copies of itself.

## Proof 1

By the tableau method of natural deduction:

$\vdash p \iff p$
Line Pool Formula Rule Depends upon Notes
1 $p \implies p$ Theorem Introduction (None) Law of Identity: Formulation 2
2 $p \iff p$ Biconditional Introduction: $\iff \II$ 1, 1

$\blacksquare$

## Proof 2

We apply the Method of Truth Tables.

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

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

$\blacksquare$