Rule of Commutation/Disjunction/Formulation 2/Proof 1

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Theorem

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


Proof

By the tableau method of natural deduction:

$\vdash \left({p \lor q}\right) \iff \left({q \lor p}\right)$
Line Pool Formula Rule Depends upon Notes
1 1 $p \lor q$ Assumption (None)
2 1 $q \lor p$ Sequent Introduction 1 Disjunction is Commutative
3 $\left({p \lor q}\right) \implies \left({q \lor p}\right)$ Rule of Implication: $\implies \mathcal I$ 1 – 2 Assumption 1 has been discharged
4 4 $q \lor p$ Assumption (None)
5 4 $p \lor q$ Sequent Introduction 4 Disjunction is Commutative
6 $\left({q \lor p}\right) \implies \left({p \lor q}\right)$ Rule of Implication: $\implies \mathcal I$ 4 – 5 Assumption 4 has been discharged
7 $\left({p \lor q}\right) \iff \left({q \lor p}\right)$ Biconditional Introduction: $\iff \mathcal I$ 3, 6

$\blacksquare$