Theorem pm2.61dan 766

Description: Elimination of an antecedent. (Contributed by NM, 1-Jan-2005.)

Hypotheses

Ref	Expression
pm2.61dan.1	⊢ ((φ ∧ ψ) → χ)
pm2.61dan.2	⊢ ((φ ∧ ¬ ψ) → χ)

Assertion

Ref	Expression
pm2.61dan	⊢ (φ → χ)

Proof of Theorem pm2.61dan

Step	Hyp	Ref	Expression
1		pm2.61dan.1	. . 3 ⊢ ((φ ∧ ψ) → χ)
2	1	ex 423	. 2 ⊢ (φ → (ψ → χ))
3		pm2.61dan.2	. . 3 ⊢ ((φ ∧ ¬ ψ) → χ)
4	3	ex 423	. 2 ⊢ (φ → (¬ ψ → χ))
5	2, 4	pm2.61d 150	1 ⊢ (φ → χ)

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:  pm2.61ddan  767  pm2.61dda  768  ifeq1da  3688  ifeq2da  3689  ifclda  3690  ifbothda  3693  xpcan  5058  fvmpti  5700  fvmptss  5706