Biconditional is Transitive/Formulation 2

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The biconditional operator is transitive:

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


By the tableau method of natural deduction:

$\vdash \paren {\paren {p \iff q} \land \paren {q \iff r} } \implies \paren {p \iff r} $
Line Pool Formula Rule Depends upon Notes
1 1 $\paren {p \iff q} \land \paren {q \iff r}$ Assumption (None)
2 1 $p \iff q$ Rule of Simplification: $\land \EE_1$ 1
3 1 $q \iff r$ Rule of Simplification: $\land \EE_2$ 1
4 1 $p \iff r$ Sequent Introduction 2, 3 Biconditional is Transitive: Formulation 1
5 $\paren {\paren {p \iff q} \land \paren {q \iff r} } \implies \paren {p \iff r}$ Rule of Implication: $\implies \II$ 1 – 4 Assumption 1 has been discharged