Theorem anabs7 677

Description: Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 17-Nov-2013.)

Assertion

Ref	Expression
anabs7	⊢ ((𝜓 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ 𝜓))

Proof of Theorem anabs7

Step	Hyp	Ref	Expression
1		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
2	1	pm4.71ri 570	. 2 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ (𝜑 ∧ 𝜓)))
3	2	bicomi 227	1 ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜓)) ↔ (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:  prtlem15  39912  un2122  45757