# Rule of Transposition/Variant 1/Formulation 1/Proof 1

## Theorem

$p \implies \neg q \dashv \vdash q \implies \neg p$

## Proof

### Proof of Forward Implication

By the tableau method of natural deduction:

$p \implies \neg q \vdash q \implies \neg p$
Line Pool Formula Rule Depends upon Notes
1 1 $p \implies \neg q$ Premise (None)
2 2 $q$ Assumption (None)
3 2 $\neg \neg q$ Double Negation Introduction: $\neg \neg \II$ 2
4 1, 2 $\neg p$ Modus Tollendo Tollens (MTT) 1, 3
5 1 $q \implies \neg p$ Rule of Implication: $\implies \II$ 2 – 4 Assumption 2 has been discharged

$\blacksquare$

### Proof of Reverse Implication

By the tableau method of natural deduction:

$q \implies \neg p \vdash p \implies \neg q$
Line Pool Formula Rule Depends upon Notes
1 1 $q \implies \neg p$ Premise (None)
2 2 $p$ Assumption (None)
3 2 $\neg \neg p$ Double Negation Introduction: $\neg \neg \II$ 2
4 1, 2 $\neg q$ Modus Tollendo Tollens (MTT) 1, 3
5 1 $p \implies \neg q$ Rule of Implication: $\implies \II$ 2 – 4 Assumption 2 has been discharged

$\blacksquare$