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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  |-  ( ( ( ph  \/  ps )  ->  ( ph  <->  ps )
)  <->  ( ph  <->  ps )
)

Proof of Theorem oibabs
StepHypRef Expression
1 ioran 476 . . . 4  |-  ( -.  ( ph  \/  ps ) 
<->  ( -.  ph  /\  -.  ps ) )
2 pm5.21 831 . . . 4  |-  ( ( -.  ph  /\  -.  ps )  ->  ( ph  <->  ps )
)
31, 2sylbi 187 . . 3  |-  ( -.  ( ph  \/  ps )  ->  ( ph  <->  ps )
)
4 id 19 . . 3  |-  ( (
ph 
<->  ps )  ->  ( ph 
<->  ps ) )
53, 4ja 153 . 2  |-  ( ( ( ph  \/  ps )  ->  ( ph  <->  ps )
)  ->  ( ph  <->  ps ) )
6 ax-1 5 . 2  |-  ( (
ph 
<->  ps )  ->  (
( ph  \/  ps )  ->  ( ph  <->  ps )
) )
75, 6impbii 180 1  |-  ( ( ( ph  \/  ps )  ->  ( ph  <->  ps )
)  <->  ( ph  <->  ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360
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