Theorem dcnn 855

Description: Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 849. The relation between dcn 849 and dcnn 855 is analogous to that between notnot 634 and notnotnot 639 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 855 means that a proposition is testable if and only if its negation is testable, and dcn 849 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 849	. 2 |- (DECID -. ph -> DECID -. -. ph )
2		stabnot 840	. . 3 |- STAB -. ph
3		stdcn 854	. . 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 837  DECID wdc 841

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 619  ax-in2 620  ax-io 716

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842

This theorem is referenced by: (None)