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Theorem stabtestimpdc 862
Description: "Stable and testable" is equivalent to decidable. (Contributed by David A. Wheeler, 13-Aug-2018.)
Assertion
Ref Expression
stabtestimpdc  |-  ( (STAB  ph  /\ DECID  -.  ph )  <-> DECID  ph )

Proof of Theorem stabtestimpdc
StepHypRef Expression
1 exmiddc 782 . . . . . 6  |-  (DECID  -.  ph  ->  ( -.  ph  \/  -.  -.  ph ) )
21adantl 271 . . . . 5  |-  ( (STAB  ph  /\ DECID  -.  ph )  ->  ( -.  ph  \/  -.  -.  ph ) )
3 df-stab 776 . . . . . . . 8  |-  (STAB  ph  <->  ( -.  -.  ph  ->  ph ) )
43biimpi 118 . . . . . . 7  |-  (STAB  ph  ->  ( -.  -.  ph  ->  ph ) )
54orim2d 737 . . . . . 6  |-  (STAB  ph  ->  ( ( -.  ph  \/  -.  -.  ph )  -> 
( -.  ph  \/  ph ) ) )
65adantr 270 . . . . 5  |-  ( (STAB  ph  /\ DECID  -.  ph )  ->  ( ( -.  ph  \/  -.  -.  ph )  ->  ( -.  ph  \/  ph ) ) )
72, 6mpd 13 . . . 4  |-  ( (STAB  ph  /\ DECID  -.  ph )  ->  ( -.  ph  \/  ph ) )
87orcomd 683 . . 3  |-  ( (STAB  ph  /\ DECID  -.  ph )  ->  ( ph  \/  -.  ph ) )
9 df-dc 781 . . 3  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
108, 9sylibr 132 . 2  |-  ( (STAB  ph  /\ DECID  -.  ph )  -> DECID  ph )
11 dcimpstab 790 . . 3  |-  (DECID  ph  -> STAB  ph )
12 dcn 784 . . 3  |-  (DECID  ph  -> DECID  -.  ph )
1311, 12jca 300 . 2  |-  (DECID  ph  ->  (STAB  ph  /\ DECID  -.  ph ) )
1410, 13impbii 124 1  |-  ( (STAB  ph  /\ DECID  -.  ph )  <-> DECID  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664  STAB wstab 775  DECID wdc 780
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  ax-io 665
This theorem depends on definitions:  df-bi 115  df-stab 776  df-dc 781
This theorem is referenced by: (None)
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