Theorem anabsi5 670

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 207  df-an 396

This theorem is referenced by:  anabsi6  671  anabsi8  673  3anidm12  1422  rspce  3554  onint  7746  f1oweALT  7927  hasheqf1oi  14315  rtrclreclem3  15024  rtrclreclem4  15025  ablsimpgfindlem1  20086  ptcmpfi  23780  redwlk  29741  frgruhgr0v  30336  finxpreclem2  37708  finxpreclem6  37714  diophin  43206  diophun  43207  rspcegf  45456  stoweidlem36  46466  grlimgrtri  48481