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Theorem oibabs 686
Description: Absorption of disjunction into equivalence. (Contributed by NM, 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.)
Assertion
Ref Expression
oibabs (((𝜑𝜓) → (𝜑𝜓)) ↔ (𝜑𝜓))

Proof of Theorem oibabs
StepHypRef Expression
1 pm2.67-2 685 . . . 4 (((𝜑𝜓) → (𝜑𝜓)) → (𝜑 → (𝜑𝜓)))
21ibd 177 . . 3 (((𝜑𝜓) → (𝜑𝜓)) → (𝜑𝜓))
3 olc 683 . . . . 5 (𝜓 → (𝜑𝜓))
43imim1i 60 . . . 4 (((𝜑𝜓) → (𝜑𝜓)) → (𝜓 → (𝜑𝜓)))
5 ibibr 245 . . . 4 ((𝜓𝜑) ↔ (𝜓 → (𝜑𝜓)))
64, 5sylibr 133 . . 3 (((𝜑𝜓) → (𝜑𝜓)) → (𝜓𝜑))
72, 6impbid 128 . 2 (((𝜑𝜓) → (𝜑𝜓)) → (𝜑𝜓))
8 ax-1 6 . 2 ((𝜑𝜓) → ((𝜑𝜓) → (𝜑𝜓)))
97, 8impbii 125 1 (((𝜑𝜓) → (𝜑𝜓)) ↔ (𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wo 680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681
This theorem depends on definitions:  df-bi 116
This theorem is referenced by: (None)
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