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

Theorem dcimpstab 791
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 790 . 2 (DECID 𝜑 → (¬ ¬ 𝜑𝜑))
2 df-stab 777 . 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 776  DECID wdc 781
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 581  ax-io 666
This theorem depends on definitions:  df-bi 116  df-stab 777  df-dc 782
This theorem is referenced by:  stabtestimpdc  863  sbthlemi3  6722
  Copyright terms: Public domain W3C validator