Theorem anabss5 665

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 468	. 2 ⊢ (((𝜑 ∧ 𝜑) ∧ 𝜓) → 𝜒)
3	2	anabsan 662	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 396

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 397

This theorem is referenced by:  mp3an2ani  1467  sq01  13940  hashssdif  14127  eqbrrdv2  36877  expgrowthi  41951  bccbc  41963  hoidmvlelem2  44134