Rule of Transposition/Variant 1/Formulation 1/Reverse Implication

Theorem

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

Proof

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

$\blacksquare$