Theorem anabsan 565

Description: Absorption of antecedent with conjunction. (Contributed by NM, 24-Mar-1996.) (Revised by NM, 18-Nov-2013.)

Hypothesis

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

Assertion

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

Proof of Theorem anabsan

Step	Hyp	Ref	Expression
1		pm4.24 393	. 2 ⊢ (𝜑 ↔ (𝜑 ∧ 𝜑))
2		anabsan.1	. 2 ⊢ (((𝜑 ∧ 𝜑) ∧ 𝜓) → 𝜒)
3	1, 2	sylanb 282	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  anabss1  566  anabss5  568  anandis  582  iddvds  11744  1dvds  11745