Biconditional Elimination/Sequent Form

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Theorem

$(1): \quad p \iff q \vdash p \implies q$
$(2): \quad p \iff q \vdash q \implies p$


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: $\iff \mathcal E_1$ 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: $\iff \mathcal E_2$ 1

$\blacksquare$


Proof 2

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 $q \implies p$.

$\blacksquare$


Sources