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

Theorem dcstab 829
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 828 . 2 (DECID 𝜑 → (¬ ¬ 𝜑𝜑))
2 df-stab 816 . 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 815  DECID wdc 819
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 604  ax-io 698
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820
This theorem is referenced by:  stdcndc  830  stdcndcOLD  831  condc  838  imandc  874  sbthlemi3  6847  bj-nnst  12994
  Copyright terms: Public domain W3C validator