Biconditional is Commutative/Formulation 2

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Theorems

The biconditional operator is commutative:

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


Proof

By the tableau method of natural deduction:

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

$\blacksquare$


Sources