# Biconditional Elimination/Sequent Form

## Theorem

 $\text {(1)}: \quad$ $\ds p \iff q$ $\vdash$ $\ds p \implies q$ $\text {(2)}: \quad$ $\ds p \iff q$ $\vdash$ $\ds q \implies p$

## Proof 1

### Form 1

By the tableau method of natural deduction:

$p \iff q \vdash p \implies q$
Line Pool Formula Rule Depends upon Notes
1 1 $p \iff q$ Premise (None)
2 1 $p \implies q$ Biconditional Elimination: $\iff \EE_1$ 1

$\blacksquare$

### Form 2

By the tableau method of natural deduction:

$p \iff q \vdash q \implies p$
Line Pool Formula Rule Depends upon Notes
1 1 $p \iff q$ Premise (None)
2 1 $q \implies p$ Biconditional Elimination: $\iff \EE_2$ 1

$\blacksquare$

## Proof by Truth Table

We apply the Method of Truth Tables.

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

As can be seen, when $p \iff q$ is true so are both $p \implies q$ and $q \implies p$.

$\blacksquare$