Theorem stdcndc 846

Description: A formula is decidable if and only if its negation is decidable and it is stable (that is, it is testable and stable). (Contributed by David A. Wheeler, 13-Aug-2018.) (Proof shortened by BJ, 28-Oct-2023.)

Assertion

Ref	Expression
stdcndc	|- ( (STAB ph /\ DECID -. ph ) <-> DECID ph )

Proof of Theorem stdcndc

Step	Hyp	Ref	Expression
1		df-stab 832	. . . 4 |- (STAB ph <-> ( -. -. ph -> ph ) )
2		df-dc 836	. . . 4 |- (DECID -. ph <-> ( -. ph \/ -. -. ph ) )
3		pm2.36 805	. . . . 5 |- ( ( -. -. ph -> ph ) -> ( ( -. ph \/ -. -. ph ) -> ( ph \/ -. ph ) ) )
4	3	imp 124	. . . 4 |- ( ( ( -. -. ph -> ph ) /\ ( -. ph \/ -. -. ph ) ) -> ( ph \/ -. ph ) )
5	1, 2, 4	syl2anb 291	. . 3 |- ( (STAB ph /\ DECID -. ph ) -> ( ph \/ -. ph ) )
6		df-dc 836	. . 3 |- (DECID ph <-> ( ph \/ -. ph ) )
7	5, 6	sylibr 134	. 2 |- ( (STAB ph /\ DECID -. ph ) -> DECID ph )
8		dcstab 845	. . 3 |- (DECID ph -> STAB ph )
9		dcn 843	. . 3 |- (DECID ph -> DECID -. ph )
10	8, 9	jca 306	. 2 |- (DECID ph -> (STAB ph /\ DECID -. ph ) )
11	7, 10	impbii 126	1 |- ( (STAB ph /\ DECID -. ph ) <-> DECID ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  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-in1 615  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:  stdcn  848