ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  oibabs Unicode version

Theorem oibabs 834
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 pm2.67-2 667 . . . 4  |-  ( ( ( ph  \/  ps )  ->  ( ph  <->  ps )
)  ->  ( ph  ->  ( ph  <->  ps )
) )
21ibd 176 . . 3  |-  ( ( ( ph  \/  ps )  ->  ( ph  <->  ps )
)  ->  ( ph  ->  ps ) )
3 olc 665 . . . . 5  |-  ( ps 
->  ( ph  \/  ps ) )
43imim1i 59 . . . 4  |-  ( ( ( ph  \/  ps )  ->  ( ph  <->  ps )
)  ->  ( ps  ->  ( ph  <->  ps )
) )
5 ibibr 244 . . . 4  |-  ( ( ps  ->  ph )  <->  ( ps  ->  ( ph  <->  ps )
) )
64, 5sylibr 132 . . 3  |-  ( ( ( ph  \/  ps )  ->  ( ph  <->  ps )
)  ->  ( ps  ->  ph ) )
72, 6impbid 127 . 2  |-  ( ( ( ph  \/  ps )  ->  ( ph  <->  ps )
)  ->  ( ph  <->  ps ) )
8 ax-1 5 . 2  |-  ( (
ph 
<->  ps )  ->  (
( ph  \/  ps )  ->  ( ph  <->  ps )
) )
97, 8impbii 124 1  |-  ( ( ( ph  \/  ps )  ->  ( ph  <->  ps )
)  <->  ( ph  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    \/ wo 662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator