# Biconditional is Associative/Formulation 2

## Theorem

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

## Proof

By the tableau method of natural deduction:

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

$\blacksquare$