Theorem oranabs 728

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 664	. . 3 ⊢ (𝜓 → (𝜑 ↔ (¬ 𝜓 ∨ 𝜑)))
2		orcom 647	. . 3 ⊢ ((¬ 𝜓 ∨ 𝜑) ↔ (𝜑 ∨ ¬ 𝜓))
3	1, 2	syl6rbb 186	. 2 ⊢ (𝜓 → ((𝜑 ∨ ¬ 𝜓) ↔ 𝜑))
4	3	pm5.32ri 428	1 ⊢ (((𝜑 ∨ ¬ 𝜓) ∧ 𝜓) ↔ (𝜑 ∧ 𝜓))

Syntax hints:  ¬ wn 3   ∧ wa 97   ↔ wb 98   ∨ wo 629

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)