Theorem bj-stdc 16082

Description: Decidability of a proposition is stable if and only if that proposition is decidable. In particular, the assumption that every formula is stable implies that every formula is decidable, hence classical logic. (Contributed by BJ, 9-Oct-2019.)

Assertion

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

Proof of Theorem bj-stdc

Step	Hyp	Ref	Expression
1		nndc 856	. 2 ⊢ ¬ ¬ DECID 𝜑
2		bj-nnbist 16066	. 2 ⊢ (¬ ¬ DECID 𝜑 → (STAB DECID 𝜑 ↔ DECID 𝜑))
3	1, 2	ax-mp 5	1 ⊢ (STAB DECID 𝜑 ↔ DECID 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 105  STAB wstab 835  DECID wdc 839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840

This theorem is referenced by: (None)