Theorem bj-nnbidc 13792

Description: If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 13779. (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 13791	. . 3 |- (DECID ph -> STAB ph )
2		bj-nnbist 13779	. . 3 |- ( -. -. ph -> (STAB ph <-> ph ) )
3	1, 2	syl5ib 153	. 2 |- ( -. -. ph -> (DECID ph -> ph ) )
4		bj-trdc 13787	. 2 |- ( ph -> DECID ph )
5	3, 4	impbid1 141	1 |- ( -. -. ph -> (DECID ph <-> ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104  STAB wstab 825  DECID wdc 829

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 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830

This theorem is referenced by:  bj-dcdc  13794  bj-dcst  13796