Theorem bj-stdc 16356

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 858	. 2 |- -. -. DECID ph
2		bj-nnbist 16340	. 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 105  STAB wstab 837  DECID wdc 841

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 619  ax-in2 620  ax-io 716

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842

This theorem is referenced by: (None)