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Theorem oibabs 851
Description: Absorption of disjunction into equivalence. (Contributed by NM, 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.)
Assertion
Ref Expression
oibabs (((φ ψ) → (φψ)) ↔ (φψ))

Proof of Theorem oibabs
StepHypRef Expression
1 ioran 476 . . . 4 (¬ (φ ψ) ↔ (¬ φ ¬ ψ))
2 pm5.21 831 . . . 4 ((¬ φ ¬ ψ) → (φψ))
31, 2sylbi 187 . . 3 (¬ (φ ψ) → (φψ))
4 id 19 . . 3 ((φψ) → (φψ))
53, 4ja 153 . 2 (((φ ψ) → (φψ)) → (φψ))
6 ax-1 6 . 2 ((φψ) → ((φ ψ) → (φψ)))
75, 6impbii 180 1 (((φ ψ) → (φψ)) ↔ (φψ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  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)
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