Theorem bj-stdc 12966

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 ph <-> DECID ph )

Proof of Theorem bj-stdc

Step	Hyp	Ref	Expression
1		nndc 836	. 2 |- -. -. DECID ph
2		bj-nnbist 12953	. 2 |- ( -. -. DECID ph -> (STAB DECID ph <-> DECID ph ) )
3	1, 2	ax-mp 5	1 |- (STAB DECID ph <-> DECID ph )

Syntax hints:   -. wn 3   <-> wb 104  STAB wstab 815  DECID wdc 819

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 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820

This theorem is referenced by: (None)