Theorem mtand 665

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 115	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
4	1, 3	mtod 663	1 ⊢ (𝜑 → ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-in1 614  ax-in2 615

This theorem is referenced by:  frirrg  4351  phpm  6865  diffisn  6893  tridc  6899  nnnninfeq  7126  pm54.43  7189  addcanprleml  7613  addcanprlemu  7614  iseqf1olemklt  10485  isprm5lem  12141  pw2dvdseulemle  12167  sqne2sq  12177  pythagtriplem4  12268  pythagtriplem11  12274  pythagtriplem13  12276  ctinfomlemom  12428  ivthinc  14124  pwle2  14751