# Modus Tollendo Tollens/Sequent Form/Proof 2

## Theorem

$p \implies q, \neg q \vdash \neg p$

## Proof

By the tableau method of natural deduction:

$p \implies q, \neg q \vdash \neg p$
Line Pool Formula Rule Depends upon Notes
1 1 $p \implies q$ Premise (None)
2 2 $\neg q$ Premise (None)
3 3 $p$ Assumption (None) Assume $p$ ...
4 1, 3 $q$ Modus Ponendo Ponens: $\implies \mathcal E$ 1, 3 ... and derive $q$ ...
5 1, 2, 3 $\bot$ Principle of Non-Contradiction: $\neg \mathcal E$ 4, 2 ... demonstrating a contradiction
6 1, 2 $\neg p$ Proof by Contradiction: $\neg \mathcal I$ 3 – 5 Assumption 3 has been discharged

$\blacksquare$