Theorem bj-dcst 13642

Description: Stability of a proposition is decidable if and only if that proposition is stable. (Contributed by BJ, 24-Nov-2023.)

Assertion

Ref	Expression
bj-dcst	⊢ (DECID STAB 𝜑 ↔ STAB 𝜑)

Proof of Theorem bj-dcst

Step	Hyp	Ref	Expression
1		bj-nnst 13624	. 2 ⊢ ¬ ¬ STAB 𝜑
2		bj-nnbidc 13638	. 2 ⊢ (¬ ¬ STAB 𝜑 → (DECID STAB 𝜑 ↔ STAB 𝜑))
3	1, 2	ax-mp 5	1 ⊢ (DECID STAB 𝜑 ↔ STAB 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 104  STAB wstab 820  DECID wdc 824

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-in1 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825

This theorem is referenced by: (None)