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Theorem dcimpstab 763
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 762 . 2 (DECID 𝜑 → (¬ ¬ 𝜑𝜑))
2 df-stab 751 . 2 (STAB 𝜑 ↔ (¬ ¬ 𝜑𝜑))
31, 2sylibr 141 1 (DECID 𝜑STAB 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  STAB wstab 750  DECID wdc 753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-stab 751  df-dc 754
This theorem is referenced by:  stabtestimpdc  835
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