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Theorem dcimpstab 786
Description: Decidability implies stability. The converse is not necessarily true. (Contributed by David A. Wheeler, 13-Aug-2018.)
Assertion
Ref Expression
dcimpstab  |-  (DECID  ph  -> STAB  ph )

Proof of Theorem dcimpstab
StepHypRef Expression
1 notnotrdc 785 . 2  |-  (DECID  ph  ->  ( -.  -.  ph  ->  ph ) )
2 df-stab 774 . 2  |-  (STAB  ph  <->  ( -.  -.  ph  ->  ph ) )
31, 2sylibr 132 1  |-  (DECID  ph  -> STAB  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4  STAB wstab 773  DECID wdc 776
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-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-stab 774  df-dc 777
This theorem is referenced by:  stabtestimpdc  858
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