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

Theorem stdcndc 831
Description: A formula is decidable if and only if its negation is decidable and it is stable (that is, it is testable and stable). (Contributed by David A. Wheeler, 13-Aug-2018.) (Proof shortened by BJ, 28-Oct-2023.)
Assertion
Ref Expression
stdcndc  |-  ( (STAB  ph  /\ DECID  -.  ph )  <-> DECID  ph )

Proof of Theorem stdcndc
StepHypRef Expression
1 df-stab 817 . . . 4  |-  (STAB  ph  <->  ( -.  -.  ph  ->  ph ) )
2 df-dc 821 . . . 4  |-  (DECID  -.  ph  <->  ( -.  ph  \/  -.  -.  ph ) )
3 pm2.36 794 . . . . 5  |-  ( ( -.  -.  ph  ->  ph )  ->  ( ( -.  ph  \/  -.  -.  ph )  ->  ( ph  \/  -.  ph ) ) )
43imp 123 . . . 4  |-  ( ( ( -.  -.  ph  ->  ph )  /\  ( -.  ph  \/  -.  -.  ph ) )  ->  ( ph  \/  -.  ph )
)
51, 2, 4syl2anb 289 . . 3  |-  ( (STAB  ph  /\ DECID  -.  ph )  ->  ( ph  \/  -.  ph ) )
6 df-dc 821 . . 3  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
75, 6sylibr 133 . 2  |-  ( (STAB  ph  /\ DECID  -.  ph )  -> DECID  ph )
8 dcstab 830 . . 3  |-  (DECID  ph  -> STAB  ph )
9 dcn 828 . . 3  |-  (DECID  ph  -> DECID  -.  ph )
108, 9jca 304 . 2  |-  (DECID  ph  ->  (STAB  ph  /\ DECID  -.  ph ) )
117, 10impbii 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:  stdcn  833
  Copyright terms: Public domain W3C validator