Biconditional Equivalent to Biconditional of Negations/Formulation 1/Forward Implication

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$p \iff q \vdash \neg p \iff \neg q$


By the tableau method of natural deduction:

$p \iff q \vdash \neg p \iff \neg q$
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
3 1 $\neg q \implies \neg p$ Sequent Introduction 2 Rule of Transposition
4 1 $q \implies p$ Biconditional Elimination: $\iff \EE_2$ 1
5 1 $\neg p \implies \neg q$ Sequent Introduction 4 Rule of Transposition
6 1 $\neg p \iff \neg q$ Biconditional Introduction: $\iff \II$ 5, 3