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  3611  onint  7810  f1oweALT  7997  php2OLD  9260  hasheqf1oi  14390  rtrclreclem3  15099  rtrclreclem4  15100  ablsimpgfindlem1  20127  ptcmpfi  23821  redwlk  29690  frgruhgr0v  30283  finxpreclem2  37391  finxpreclem6  37397  diophin  42783  diophun  42784  rspcegf  45028  stoweidlem36  46051  grlimgrtri  47963