Non-Equivalence of Proposition and Negation/Formulation 2

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Theorem

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


Proof

By the tableau method of natural deduction:

$$\vdash \neg \left({p \iff \neg p}\right)$$
Line Pool Formula Rule Depends upon Notes
1 1 $p \iff \neg p$ Assumption (None)
2 1 $p \implies \neg p$ Biconditional Elimination: $\iff \EE_1$ 1
3 1 $\neg p \implies p$ Biconditional Elimination: $\iff \EE_2$ 1
4 1 $\bot$ Sequent Introduction 2,3 Non-Equivalence of Proposition and Negation: Formulation 1
5 $\neg \left({p \iff \neg p}\right)$ Proof by Contradiction: $\neg \II$ 1 – 4 Assumption 1 has been discharged