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Theorem dcstab 844
Description: Decidability implies stability. The converse need not hold. (Contributed by David A. Wheeler, 13-Aug-2018.)
Assertion
Ref Expression
dcstab  |-  (DECID  ph  -> STAB  ph )

Proof of Theorem dcstab
StepHypRef Expression
1 notnotrdc 843 . 2  |-  (DECID  ph  ->  ( -.  -.  ph  ->  ph ) )
2 df-stab 831 . 2  |-  (STAB  ph  <->  ( -.  -.  ph  ->  ph ) )
31, 2sylibr 134 1  |-  (DECID  ph  -> STAB  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4  STAB wstab 830  DECID wdc 834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 615  ax-io 709
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835
This theorem is referenced by:  stdcndc  845  stdcndcOLD  846  condc  853  imandc  889  sbthlemi3  6957
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