Theorem anabss5 853

Description: Absorption of antecedent into conjunction. (Contributed by NM, 10-May-1994.) (Proof shortened by Wolf Lammen, 1-Jan-2013.)

Hypothesis

Ref	Expression
anabss5.1	⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) → 𝜒)

Assertion

Ref	Expression
anabss5	⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Proof of Theorem anabss5

Step	Hyp	Ref	Expression
1		anabss5.1	. . 3 ⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) → 𝜒)
2	1	anassrs 678	. 2 ⊢ (((𝜑 ∧ 𝜑) ∧ 𝜓) → 𝜒)
3	2	anabsan 850	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 383

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 196  df-an 385

This theorem is referenced by:  anabsi5  854  syl2an2r  872  mp3an2ani  1423  sq01  12848  faclbnd5  12947  hashssdif  13061  eqbrrdv2  33166  expgrowthi  37554  bccbc  37566  hoidmvlelem2  39486