Theorem anabs1 506

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

Assertion

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

Proof of Theorem anabs1

Step	Hyp	Ref	Expression
1		simpl 102	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	pm4.71i 371	. 2 ⊢ ((𝜑 ∧ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜑))
3	2	bicomi 123	1 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜑) ↔ (𝜑 ∧ 𝜓))

Syntax hints:   ∧ wa 97   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  poirr  4044