Theorem anabsan 678

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

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		pm4.24 574	. 2 ⊢ (𝜑 ↔ (𝜑 ∧ 𝜑))
2		anabsan.1	. 2 ⊢ (((𝜑 ∧ 𝜑) ∧ 𝜓) → 𝜒)
3	1, 2	sylanb 593	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  anabss1  679  anabss5  681  anandis  691  iddvds  16344  1dvds  16345