Theorem bj-stdc 14909

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 852	. 2 ⊢ ¬ ¬ DECID 𝜑
2		bj-nnbist 14893	. 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 831  DECID wdc 835

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 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836

This theorem is referenced by: (None)