Rule of Commutation/Conjunction/Formulation 2

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Theorem

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


Proof

By the tableau method of natural deduction:

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

$\blacksquare$


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