Equivalences are Interderivable/Reverse Implication

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Theorem

If two propositional formulas are interderivable, they are equivalent:

$\left ({p \iff q}\right) \vdash \left ({p \dashv \vdash q}\right)$


Proof

By the tableau method of natural deduction:

$p \iff q \vdash \left({p \vdash q}\right)$
Line Pool Formula Rule Depends upon Notes
1 1 $p \iff q$ Premise (None)
3 1 $p \implies q$ Biconditional Elimination: $\iff \EE_1$ 1
3 3 $p$ Assumption (None)
4 1, 3 $q$ Modus Ponendo Ponens: $\implies \mathcal E$ 2, 3

$\Box$


By the tableau method of natural deduction:

$p \iff q \vdash \left({q \vdash p}\right)$
Line Pool Formula Rule Depends upon Notes
1 1 $p \iff q$ Premise (None)
3 1 $p \implies q$ Biconditional Elimination: $\iff \EE_2$ 1
3 3 $q$ Assumption (None)
4 1, 3 $p$ Modus Ponendo Ponens: $\implies \mathcal E$ 2, 3

$\blacksquare$