Theorem dcstab 840

Description: Decidability implies stability. The converse need not hold. (Contributed by David A. Wheeler, 13-Aug-2018.)

Assertion

Ref	Expression
dcstab	⊢ (DECID 𝜑 → STAB 𝜑)

Proof of Theorem dcstab

Step	Hyp	Ref	Expression
1		notnotrdc 839	. 2 ⊢ (DECID 𝜑 → (¬ ¬ 𝜑 → 𝜑))
2		df-stab 827	. 2 ⊢ (STAB 𝜑 ↔ (¬ ¬ 𝜑 → 𝜑))
3	1, 2	sylibr 133	1 ⊢ (DECID 𝜑 → STAB 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  STAB wstab 826  DECID wdc 830

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 611  ax-io 705

This theorem depends on definitions:  df-bi 116  df-stab 827  df-dc 831

This theorem is referenced by:  stdcndc  841  stdcndcOLD  842  condc  849  imandc  885  sbthlemi3  6940