Metamath Proof Explorer

Theorem anabs1

Description: Absorption into embedded conjunct. (Contributed by NM, 4-Sep-1995) (Proof shortened by Wolf Lammen, 16-Nov-2013)

Ref Expression
Assertion anabs1 ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜑 ) ↔ ( 𝜑 ∧ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 simpl ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜑 )
2 1 pm4.71i ⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜑 ) )
3 2 bicomi ⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜑 ) ↔ ( 𝜑 ∧ 𝜓 ) )