# Rule of Transposition/Formulation 1/Forward Implication/Proof

## Theorem

 $\ds p$ $\implies$ $\ds q$ $\ds \vdash \ \$ $\ds \neg q$ $\implies$ $\ds \neg p$

## Proof

By the tableau method of natural deduction:

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

$\blacksquare$