Theorem mto

Description: The rule of modus tollens. The rule says, "if ps is not true, and ph implies ps , then ph must also be not true". Modus tollens is short for "modus tollendo tollens", a Latin phrase that means "the mode that by denying denies" - remark in Sanford p. 39. It is also called denying the consequent. Modus tollens is closely related to modus ponens ax-mp . Note that this rule is also valid in intuitionistic logic. Inference associated with con3i . (Contributed by NM, 19-Aug-1993) (Proof shortened by Wolf Lammen, 11-Sep-2013)

	Ref	Expression
Hypotheses	mto.1	$$⊢\neg \psi$$
	mto.2	$$⊢\phi \to \psi$$
Assertion	mto	$$⊢\neg \phi$$

Proof

Step	Hyp	Ref	Expression
1		mto.1	$$⊢\neg \psi$$
2		mto.2	$$⊢\phi \to \psi$$
3	1	a1i	$$⊢\phi \to \neg \psi$$
4	2 3	pm2.65i	$$⊢\neg \phi$$