Theorem oranabs 829

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 395	. . 3 ⊢ (ψ → (φ ↔ (¬ ψ ∨ φ)))
2		orcom 376	. . 3 ⊢ ((¬ ψ ∨ φ) ↔ (φ ∨ ¬ ψ))
3	1, 2	syl6rbb 253	. 2 ⊢ (ψ → ((φ ∨ ¬ ψ) ↔ φ))
4	3	pm5.32ri 619	1 ⊢ (((φ ∨ ¬ ψ) ∧ ψ) ↔ (φ ∧ ψ))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358

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 177  df-or 359  df-an 360

This theorem is referenced by: (None)