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