Theorem anabs1 675

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 488	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	pm4.71i 569	. 2 ⊢ ((𝜑 ∧ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜑))
3	2	bicomi 227	1 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜑) ↔ (𝜑 ∧ 𝜓))

This proof depends on syntax axioms:   ↔ wb 209   ∧ 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:  3anidm  1121  poirr  5583  frgr3v  30697  uun121p1  45550