Theorem oranabs 897

Description: Absorb a disjunct into a conjunct. (Contributed by Roy F. Longton, 23-Jun-2005.) (Proof shortened by Wolf Lammen, 10-Nov-2013.)

Assertion

Ref	Expression
oranabs	⊢ (((𝜑 ∨ ¬ 𝜓) ∧ 𝜓) ↔ (𝜑 ∧ 𝜓))

Proof of Theorem oranabs

Step	Hyp	Ref	Expression
1		biortn 420	. . 3 ⊢ (𝜓 → (𝜑 ↔ (¬ 𝜓 ∨ 𝜑)))
2		orcom 401	. . 3 ⊢ ((¬ 𝜓 ∨ 𝜑) ↔ (𝜑 ∨ ¬ 𝜓))
3	1, 2	syl6rbb 276	. 2 ⊢ (𝜓 → ((𝜑 ∨ ¬ 𝜓) ↔ 𝜑))
4	3	pm5.32ri 668	1 ⊢ (((𝜑 ∨ ¬ 𝜓) ∧ 𝜓) ↔ (𝜑 ∧ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 195   ∨ wo 382   ∧ wa 383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385

This theorem is referenced by:  itg2addnclem3  32633