Theorem bj-dcst 16789

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 16771	. 2 ⊢ ¬ ¬ STAB 𝜑
2		bj-nnbidc 16785	. 2 ⊢ (¬ ¬ STAB 𝜑 → (DECID STAB 𝜑 ↔ STAB 𝜑))
3	1, 2	ax-mp 5	1 ⊢ (DECID STAB 𝜑 ↔ STAB 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 105  STAB wstab 842  DECID wdc 846

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847

This theorem is used by: (None)