Non-Equivalence as Disjunction of Conjunctions/Formulation 2

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Theorem

$\vdash \paren {\neg \paren {p \iff q} } \iff \paren {\paren {\neg p \land q} \lor \paren {p \land \neg q} }$


Proof

By the tableau method of natural deduction:

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

$\blacksquare$


Sources