Theorem anabss1 672

Description: Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)

Hypothesis

Ref	Expression
anabss1.1	⊢ (((𝜑 ∧ 𝜓) ∧ 𝜑) → 𝜒)

Assertion

Ref	Expression
anabss1	⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Proof of Theorem anabss1

Step	Hyp	Ref	Expression
1		anabss1.1	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜑) → 𝜒)
2	1	an32s 658	. 2 ⊢ (((𝜑 ∧ 𝜑) ∧ 𝜓) → 𝜒)
3	2	anabsan 671	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 208  df-an 397

This theorem is referenced by:  anabss4  673  ordtri3or  6342  onfununi  8271  omordi  8491  oeoelem  8524  fzindd  12622  hashssdif  14365  nzss  44761  stirlinglem5  46521