Theorem bj-stdc 13641

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 841	. 2 ⊢ ¬ ¬ DECID 𝜑
2		bj-nnbist 13625	. 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 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)