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Theorem dcstab 839
Description: Decidability implies stability. The converse need not hold. (Contributed by David A. Wheeler, 13-Aug-2018.)
Assertion
Ref Expression
dcstab (DECID 𝜑STAB 𝜑)

Proof of Theorem dcstab
StepHypRef Expression
1 notnotrdc 838 . 2 (DECID 𝜑 → (¬ ¬ 𝜑𝜑))
2 df-stab 826 . 2 (STAB 𝜑 ↔ (¬ ¬ 𝜑𝜑))
31, 2sylibr 133 1 (DECID 𝜑STAB 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  STAB wstab 825  DECID wdc 829
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-in2 610  ax-io 704
This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830
This theorem is referenced by:  stdcndc  840  stdcndcOLD  841  condc  848  imandc  884  sbthlemi3  6932
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