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Theorem stdcn 833
Description: A formula is stable if and only if the decidability of its negation implies its decidability. Note that the right-hand side of this biconditional is the converse of dcn 828. (Contributed by BJ, 18-Nov-2023.)
Assertion
Ref Expression
stdcn  |-  (STAB  ph  <->  (DECID  -.  ph  -> DECID  ph )
)

Proof of Theorem stdcn
StepHypRef Expression
1 stdcndc 831 . . . 4  |-  ( (STAB  ph  /\ DECID  -.  ph )  <-> DECID  ph )
21biimpi 119 . . 3  |-  ( (STAB  ph  /\ DECID  -.  ph )  -> DECID  ph )
32ex 114 . 2  |-  (STAB  ph  ->  (DECID  -. 
ph  -> DECID  ph ) )
4 olc 701 . . . . 5  |-  ( -. 
-.  ph  ->  ( -. 
ph  \/  -.  -.  ph ) )
54imim1i 60 . . . 4  |-  ( ( ( -.  ph  \/  -.  -.  ph )  -> 
( ph  \/  -.  ph ) )  ->  ( -.  -.  ph  ->  ( ph  \/  -.  ph ) ) )
6 orel2 716 . . . 4  |-  ( -. 
-.  ph  ->  ( (
ph  \/  -.  ph )  ->  ph ) )
75, 6sylcom 28 . . 3  |-  ( ( ( -.  ph  \/  -.  -.  ph )  -> 
( ph  \/  -.  ph ) )  ->  ( -.  -.  ph  ->  ph )
)
8 df-dc 821 . . . 4  |-  (DECID  -.  ph  <->  ( -.  ph  \/  -.  -.  ph ) )
9 df-dc 821 . . . 4  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
108, 9imbi12i 238 . . 3  |-  ( (DECID  -. 
ph  -> DECID  ph )  <->  ( ( -. 
ph  \/  -.  -.  ph )  ->  ( ph  \/  -.  ph ) ) )
11 df-stab 817 . . 3  |-  (STAB  ph  <->  ( -.  -.  ph  ->  ph ) )
127, 10, 113imtr4i 200 . 2  |-  ( (DECID  -. 
ph  -> DECID  ph )  -> STAB  ph )
133, 12impbii 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 698  STAB wstab 816  DECID wdc 820
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 604  ax-in2 605  ax-io 699
This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821
This theorem is referenced by:  dcnn  834  bj-charfunbi  13397
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