Non-Equivalence as Equivalence with Negation/Formulation 2

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Theorem

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


Proof

By the tableau method of natural deduction:

$\vdash \neg \left ({p \iff q}\right) \iff \left({p \iff \neg q}\right)$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg \left ({p \iff q}\right)$ Assumption (None)
2 1 $p \iff \neg q$ Sequent Introduction 1 Non-Equivalence as Equivalence with Negation: Formulation 1
3 $\left({\neg \left ({p \iff q}\right)}\right) \implies \left({p \iff \neg q}\right)$ Rule of Implication: $\implies \mathcal I$ 1 – 2 Assumption 1 has been discharged
4 4 $p \iff \neg q$ Assumption (None)
5 4 $\neg \left ({p \iff q}\right)$ Sequent Introduction 4 Non-Equivalence as Equivalence with Negation: Formulation 1
6 $\left({p \iff \neg q}\right) \implies \left({\neg \left ({p \iff q}\right)}\right)$ Rule of Implication: $\implies \mathcal I$ 4 – 5 Assumption 4 has been discharged
7 $\left({\neg \left ({p \iff q}\right)}\right) \iff \left({p \iff \neg q}\right)$ Biconditional Introduction: $\iff \mathcal I$ 3, 6

$\blacksquare$


Sources