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 𝜑 → STAB 𝜑)

Proof of Theorem dcimpstab

Step	Hyp	Ref	Expression
1		notnotrdc 792	. 2 ⊢ (DECID 𝜑 → (¬ ¬ 𝜑 → 𝜑))
2		df-stab 779	. 2 ⊢ (STAB 𝜑 ↔ (¬ ¬ 𝜑 → 𝜑))
3	1, 2	sylibr 133	1 ⊢ (DECID 𝜑 → STAB 𝜑)

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