Theorem mtand 623

Description: A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)

Hypotheses

Ref	Expression
mtand.1	⊢ (𝜑 → ¬ 𝜒)
mtand.2	⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Assertion

Ref	Expression
mtand	⊢ (𝜑 → ¬ 𝜓)

Proof of Theorem mtand

Step	Hyp	Ref	Expression
1		mtand.1	. 2 ⊢ (𝜑 → ¬ 𝜒)
2		mtand.2	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
3	2	ex 113	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
4	1, 3	mtod 621	1 ⊢ (𝜑 → ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia3 106  ax-in1 576  ax-in2 577

This theorem is referenced by:  frirrg  4105  phpm  6351  diffisn  6377  pm54.43  6459  addcanprleml  6804  addcanprlemu  6805  pw2dvdseulemle  10545  sqne2sq  10555