Theorem mtand 637

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 114	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
4	1, 3	mtod 635	1 ⊢ (𝜑 → ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia3 107  ax-in1 586  ax-in2 587

This theorem is referenced by:  frirrg  4230  phpm  6709  diffisn  6737  tridc  6743  pm54.43  6992  addcanprleml  7363  addcanprlemu  7364  iseqf1olemklt  10144  pw2dvdseulemle  11683  sqne2sq  11693  ctinfomlemom  11778  pwle2  12872  nninfalllemn  12879