# Biconditional Equivalent to Biconditional of Negations/Formulation 2

## Theorem

$\vdash \left({p \iff q}\right) \iff \left({\neg p \iff \neg q}\right)$

By the tableau method of natural deduction:

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

$\blacksquare$