Theorem dcnn 860

Description: Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 854. The relation between dcn 854 and dcnn 860 is analogous to that between notnot 638 and notnotnot 643 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 860 means that a proposition is testable if and only if its negation is testable, and dcn 854 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 854	. 2 |- (DECID -. ph -> DECID -. -. ph )
2		stabnot 845	. . 3 |- STAB -. ph
3		stdcn 859	. . 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 842  DECID wdc 846

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 623  ax-in2 624  ax-io 721

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847

This theorem is referenced by: (None)