Theorem bj-nnbidc 12965

Description: If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 12956. (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 12964	. . 3 |- (DECID ph -> STAB ph )
2		bj-nnbist 12956	. . 3 |- ( -. -. ph -> (STAB ph <-> ph ) )
3	1, 2	syl5ib 153	. 2 |- ( -. -. ph -> (DECID ph -> ph ) )
4		bj-trdc 12962	. 2 |- ( ph -> DECID ph )
5	3, 4	impbid1 141	1 |- ( -. -. ph -> (DECID ph <-> ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> 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-in2 604  ax-io 698

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

This theorem is referenced by:  bj-dcdc  12968  bj-dcst  12970