Rule of Transposition/Formulation 2/Forward Implication/Proof

Theorem

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

Proof

By the tableau method of natural deduction:

$\vdash \left({p \implies q}\right) \implies \left({\neg q \implies \neg p}\right)$
Line Pool Formula Rule Depends upon Notes
1 1 $p \implies q$ Assumption (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: $\implies \mathcal I$ 2 – 3 Assumption 2 has been discharged
5 $\left({p \implies q}\right) \implies \left({\neg q \implies \neg p}\right)$ Rule of Implication: $\implies \mathcal I$ 1 – 4 Assumption 1 has been discharged

$\blacksquare$