Theorem anabss5 666

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 470	. 2 ⊢ (((𝜑 ∧ 𝜑) ∧ 𝜓) → 𝜒)
3	2	anabsan 663	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 398

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 209  df-an 399

This theorem is referenced by:  mp3an2ani  1464  sq01  13580  hashssdif  13767  eqbrrdv2  35993  expgrowthi  40658  bccbc  40670  hoidmvlelem2  42871