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Metamath Proof Explorer

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	⊢ ¬ 𝜓
	mto.2	⊢ ( 𝜑 → 𝜓 )
Assertion	mto	⊢ ¬ 𝜑

Proof

Step	Hyp	Ref	Expression
1		mto.1	⊢ ¬ 𝜓
2		mto.2	⊢ ( 𝜑 → 𝜓 )
3	1	a1i	⊢ ( 𝜑 → ¬ 𝜓 )
4	2 3	pm2.65i	⊢ ¬ 𝜑