Theorem dcnn 848

Description: Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 842. The relation between dcn 842 and dcnn 848 is analogous to that between notnot 629 and notnotnot 634 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 848 means that a proposition is testable if and only if its negation is testable, and dcn 842 means that decidability implies testability. (Contributed by David A. Wheeler, 6-Dec-2018.) (Proof shortened by BJ, 25-Nov-2023.)

Assertion

Ref	Expression
dcnn	|- (DECID -. ph <-> DECID -. -. ph )

Proof of Theorem dcnn

Step	Hyp	Ref	Expression
1		dcn 842	. 2 |- (DECID -. ph -> DECID -. -. ph )
2		stabnot 833	. . 3 |- STAB -. ph
3		stdcn 847	. . 3 |- (STAB -. ph <-> (DECID -. -. ph -> DECID -. ph ) )
4	2, 3	mpbi 145	. 2 |- (DECID -. -. ph -> DECID -. ph )
5	1, 4	impbii 126	1 |- (DECID -. ph <-> DECID -. -. ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105  STAB wstab 830  DECID wdc 834

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 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835

This theorem is referenced by: (None)