Theorem pm2.65da 619

Description: Deduction rule for proof by contradiction. (Contributed by NM, 12-Jun-2014.)

Hypotheses

Ref	Expression
pm2.65da.1	⊢ ((𝜑 ∧ 𝜓) → 𝜒)
pm2.65da.2	⊢ ((𝜑 ∧ 𝜓) → ¬ 𝜒)

Assertion

Ref	Expression
pm2.65da	⊢ (𝜑 → ¬ 𝜓)

Proof of Theorem pm2.65da

Step	Hyp	Ref	Expression
1		pm2.65da.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
2	1	ex 113	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
3		pm2.65da.2	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝜒)
4	3	ex 113	. 2 ⊢ (𝜑 → (𝜓 → ¬ 𝜒))
5	2, 4	pm2.65d 618	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:  condandc  808  nelrdva  2797  frirrg  4105  unsnfidcex  6385  unsnfidcel  6386  prodgt0  7930  ixxdisj  8926  icodisj  9014  ltabs  9973  divalglemnqt  10320  zsupcllemstep  10341  infssuzex  10345  sqnprm  10517