MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oibabs Structured version   Visualization version   GIF version

Theorem oibabs 924
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 norbi 903 . . 3 (¬ (𝜑𝜓) → (𝜑𝜓))
2 id 22 . . 3 ((𝜑𝜓) → (𝜑𝜓))
31, 2ja 173 . 2 (((𝜑𝜓) → (𝜑𝜓)) → (𝜑𝜓))
4 ax-1 6 . 2 ((𝜑𝜓) → ((𝜑𝜓) → (𝜑𝜓)))
53, 4impbii 199 1 (((𝜑𝜓) → (𝜑𝜓)) ↔ (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 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 385
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator