Theorem oranabs 815

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 745	. . 3 ⊢ (𝜓 → (𝜑 ↔ (¬ 𝜓 ∨ 𝜑)))
2		orcom 728	. . 3 ⊢ ((¬ 𝜓 ∨ 𝜑) ↔ (𝜑 ∨ ¬ 𝜓))
3	1, 2	bitr2di 197	. 2 ⊢ (𝜓 → ((𝜑 ∨ ¬ 𝜓) ↔ 𝜑))
4	3	pm5.32ri 455	1 ⊢ (((𝜑 ∨ ¬ 𝜓) ∧ 𝜓) ↔ (𝜑 ∧ 𝜓))

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)