Theorem stdcndc 830

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 816	. . . 4 |- (STAB ph <-> ( -. -. ph -> ph ) )
2		df-dc 820	. . . 4 |- (DECID -. ph <-> ( -. ph \/ -. -. ph ) )
3		pm2.36 793	. . . . 5 |- ( ( -. -. ph -> ph ) -> ( ( -. ph \/ -. -. ph ) -> ( ph \/ -. ph ) ) )
4	3	imp 123	. . . 4 |- ( ( ( -. -. ph -> ph ) /\ ( -. ph \/ -. -. ph ) ) -> ( ph \/ -. ph ) )
5	1, 2, 4	syl2anb 289	. . 3 |- ( (STAB ph /\ DECID -. ph ) -> ( ph \/ -. ph ) )
6		df-dc 820	. . 3 |- (DECID ph <-> ( ph \/ -. ph ) )
7	5, 6	sylibr 133	. 2 |- ( (STAB ph /\ DECID -. ph ) -> DECID ph )
8		dcstab 829	. . 3 |- (DECID ph -> STAB ph )
9		dcn 827	. . 3 |- (DECID ph -> DECID -. ph )
10	8, 9	jca 304	. 2 |- (DECID ph -> (STAB ph /\ DECID -. ph ) )
11	7, 10	impbii 125	1 |- ( (STAB ph /\ DECID -. ph ) <-> DECID ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697  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-in1 603  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:  stdcn  832