Theorem bj-nnbidc 15413

Description: If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 15400. (Contributed by BJ, 24-Nov-2023.)

Assertion

Ref	Expression
bj-nnbidc	|- ( -. -. ph -> (DECID ph <-> ph ) )

Proof of Theorem bj-nnbidc

Step	Hyp	Ref	Expression
1		bj-dcstab 15412	. . 3 |- (DECID ph -> STAB ph )
2		bj-nnbist 15400	. . 3 |- ( -. -. ph -> (STAB ph <-> ph ) )
3	1, 2	imbitrid 154	. 2 |- ( -. -. ph -> (DECID ph -> ph ) )
4		bj-trdc 15408	. 2 |- ( ph -> DECID ph )
5	3, 4	impbid1 142	1 |- ( -. -. ph -> (DECID ph <-> ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> 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-in2 616  ax-io 710

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

This theorem is referenced by:  bj-dcdc  15415  bj-dcst  15417