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Theorem stdcndcOLD 814
Description: Obsolete version of stdcndc 813 as of 28-Oct-2023. (Contributed by David A. Wheeler, 13-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
stdcndcOLD  |-  ( (STAB  ph  /\ DECID  -.  ph )  <-> DECID  ph )

Proof of Theorem stdcndcOLD
StepHypRef Expression
1 exmiddc 804 . . . . . 6  |-  (DECID  -.  ph  ->  ( -.  ph  \/  -.  -.  ph ) )
21adantl 273 . . . . 5  |-  ( (STAB  ph  /\ DECID  -.  ph )  ->  ( -.  ph  \/  -.  -.  ph ) )
3 df-stab 799 . . . . . . . 8  |-  (STAB  ph  <->  ( -.  -.  ph  ->  ph ) )
43biimpi 119 . . . . . . 7  |-  (STAB  ph  ->  ( -.  -.  ph  ->  ph ) )
54orim2d 760 . . . . . 6  |-  (STAB  ph  ->  ( ( -.  ph  \/  -.  -.  ph )  -> 
( -.  ph  \/  ph ) ) )
65adantr 272 . . . . 5  |-  ( (STAB  ph  /\ DECID  -.  ph )  ->  ( ( -.  ph  \/  -.  -.  ph )  ->  ( -.  ph  \/  ph ) ) )
72, 6mpd 13 . . . 4  |-  ( (STAB  ph  /\ DECID  -.  ph )  ->  ( -.  ph  \/  ph ) )
87orcomd 701 . . 3  |-  ( (STAB  ph  /\ DECID  -.  ph )  ->  ( ph  \/  -.  ph ) )
9 df-dc 803 . . 3  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
108, 9sylibr 133 . 2  |-  ( (STAB  ph  /\ DECID  -.  ph )  -> DECID  ph )
11 dcstab 812 . . 3  |-  (DECID  ph  -> STAB  ph )
12 dcn 810 . . 3  |-  (DECID  ph  -> DECID  -.  ph )
1311, 12jca 302 . 2  |-  (DECID  ph  ->  (STAB  ph  /\ DECID  -.  ph ) )
1410, 13impbii 125 1  |-  ( (STAB  ph  /\ DECID  -.  ph )  <-> DECID  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680  STAB wstab 798  DECID wdc 802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803
This theorem is referenced by: (None)
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