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Theorem stbid 777
Description: The equivalent of a stable proposition is stable. (Contributed by Jim Kingdon, 12-Aug-2022.)
Hypothesis
Ref Expression
stbid.1  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
stbid  |-  ( ph  ->  (STAB  ps  <-> STAB  ch ) )

Proof of Theorem stbid
StepHypRef Expression
1 stbid.1 . . . . 5  |-  ( ph  ->  ( ps  <->  ch )
)
21notbid 627 . . . 4  |-  ( ph  ->  ( -.  ps  <->  -.  ch )
)
32notbid 627 . . 3  |-  ( ph  ->  ( -.  -.  ps  <->  -. 
-.  ch ) )
43, 1imbi12d 232 . 2  |-  ( ph  ->  ( ( -.  -.  ps  ->  ps )  <->  ( -.  -.  ch  ->  ch )
) )
5 df-stab 776 . 2  |-  (STAB  ps  <->  ( -.  -.  ps  ->  ps )
)
6 df-stab 776 . 2  |-  (STAB  ch  <->  ( -.  -.  ch  ->  ch )
)
74, 5, 63bitr4g 221 1  |-  ( ph  ->  (STAB  ps  <-> STAB  ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103  STAB wstab 775
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-in1 579  ax-in2 580
This theorem depends on definitions:  df-bi 115  df-stab 776
This theorem is referenced by:  dfss4st  3232
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