Metamath Proof Explorer

Theorem anabs5

Description: Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996) (Proof shortened by Wolf Lammen, 9-Dec-2012)

Ref Expression
Assertion anabs5 ( ( 𝜑 ∧ ( 𝜑 ∧ 𝜓 ) ) ↔ ( 𝜑 ∧ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 ibar ⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜑 ∧ 𝜓 ) ) )
2 1 bicomd ⊢ ( 𝜑 → ( ( 𝜑 ∧ 𝜓 ) ↔ 𝜓 ) )
3 2 pm5.32i ⊢ ( ( 𝜑 ∧ ( 𝜑 ∧ 𝜓 ) ) ↔ ( 𝜑 ∧ 𝜓 ) )