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Theorem oranabs 739
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 674 . . 3 (𝜓 → (𝜑 ↔ (¬ 𝜓𝜑)))
2 orcom 657 . . 3 ((¬ 𝜓𝜑) ↔ (𝜑 ∨ ¬ 𝜓))
31, 2syl6rbb 190 . 2 (𝜓 → ((𝜑 ∨ ¬ 𝜓) ↔ 𝜑))
43pm5.32ri 436 1 (((𝜑 ∨ ¬ 𝜓) ∧ 𝜓) ↔ (𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 101  wb 102  wo 639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114
This theorem is referenced by: (None)
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