Theorem anabsan 575

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		pm4.24 395	. 2 ⊢ (𝜑 ↔ (𝜑 ∧ 𝜑))
2		anabsan.1	. 2 ⊢ (((𝜑 ∧ 𝜑) ∧ 𝜓) → 𝜒)
3	1, 2	sylanb 284	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  anabss1  576  anabss5  578  anandis  592  iddvds  11814  1dvds  11815