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Theorem dcimpstab 793
Description: Decidability implies stability. The converse is not necessarily true. (Contributed by David A. Wheeler, 13-Aug-2018.)
Assertion
Ref Expression
dcimpstab (DECID 𝜑STAB 𝜑)

Proof of Theorem dcimpstab
StepHypRef Expression
1 notnotrdc 792 . 2 (DECID 𝜑 → (¬ ¬ 𝜑𝜑))
2 df-stab 779 . 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 778  DECID wdc 783
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 583  ax-io 668
This theorem depends on definitions:  df-bi 116  df-stab 779  df-dc 784
This theorem is referenced by:  stabtestimpdc  865  sbthlemi3  6748
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