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Theorem oranabs 937
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) ↔ (𝜑𝜓))

Proof of Theorem oranabs
StepHypRef Expression
1 biortn 420 . . 3 (𝜓 → (𝜑 ↔ (¬ 𝜓𝜑)))
2 orcom 401 . . 3 ((¬ 𝜓𝜑) ↔ (𝜑 ∨ ¬ 𝜓))
31, 2syl6rbb 277 . 2 (𝜓 → ((𝜑 ∨ ¬ 𝜓) ↔ 𝜑))
43pm5.32ri 673 1 (((𝜑 ∨ ¬ 𝜓) ∧ 𝜓) ↔ (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  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 197  df-or 384  df-an 385
This theorem is referenced by:  itg2addnclem3  33776
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