Theorem dcimpstab 785

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 784	. 2 ⊢ (DECID 𝜑 → (¬ ¬ 𝜑 → 𝜑))
2		df-stab 773	. 2 ⊢ (STAB 𝜑 ↔ (¬ ¬ 𝜑 → 𝜑))
3	1, 2	sylibr 132	1 ⊢ (DECID 𝜑 → STAB 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  STAB wstab 772  DECID wdc 775

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-stab 773  df-dc 776

This theorem is referenced by:  stabtestimpdc  857