Theorem pm2.65da 559

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 423	. 2 ⊢ (φ → (ψ → χ))
3		pm2.65da.2	. . 3 ⊢ ((φ ∧ ψ) → ¬ χ)
4	3	ex 423	. 2 ⊢ (φ → (ψ → ¬ χ))
5	2, 4	pm2.65d 166	1 ⊢ (φ → ¬ ψ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360

This theorem is referenced by:  condan  769