Theorem anabsi5 666

Description: Absorption of antecedent into conjunction. (Contributed by NM, 11-Jun-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2013.)

Hypothesis

Ref	Expression
anabsi5.1	⊢ (𝜑 → ((𝜑 ∧ 𝜓) → 𝜒))

Assertion

Ref	Expression
anabsi5	⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Proof of Theorem anabsi5

Step	Hyp	Ref	Expression
1		simpl 482	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2		anabsi5.1	. 2 ⊢ (𝜑 → ((𝜑 ∧ 𝜓) → 𝜒))
3	1, 2	mpcom 38	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 395

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 206  df-an 396

This theorem is referenced by:  anabsi6  667  anabsi8  669  3anidm12  1418  rspce  3601  onint  7782  f1oweALT  7963  php2OLD  9229  hasheqf1oi  14318  rtrclreclem3  15014  rtrclreclem4  15015  ablsimpgfindlem1  20025  ptcmpfi  23637  redwlk  29362  frgruhgr0v  29950  finxpreclem2  36735  finxpreclem6  36741  diophin  41973  diophun  41974  rspcegf  44170  stoweidlem36  45211