Theorem anabsi5 669

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  670  anabsi8  672  3anidm12  1421  rspce  3574  onint  7746  f1oweALT  7930  hasheqf1oi  14292  rtrclreclem3  15002  rtrclreclem4  15003  ablsimpgfindlem1  20015  ptcmpfi  23676  redwlk  29574  frgruhgr0v  30166  finxpreclem2  37351  finxpreclem6  37357  diophin  42733  diophun  42734  rspcegf  44990  stoweidlem36  46007  grlimgrtri  47968