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Theorem dcnn 834
 Description: Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 828. The relation between dcn 828 and dcnn 834 is analogous to that between notnot 619 and notnotnot 624 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 834 means that a proposition is testable if and only if its negation is testable, and dcn 828 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 ¬ 𝜑DECID ¬ ¬ 𝜑)

Proof of Theorem dcnn
StepHypRef Expression
1 dcn 828 . 2 (DECID ¬ 𝜑DECID ¬ ¬ 𝜑)
2 stabnot 819 . . 3 STAB ¬ 𝜑
3 stdcn 833 . . 3 (STAB ¬ 𝜑 ↔ (DECID ¬ ¬ 𝜑DECID ¬ 𝜑))
42, 3mpbi 144 . 2 (DECID ¬ ¬ 𝜑DECID ¬ 𝜑)
51, 4impbii 125 1 (DECID ¬ 𝜑DECID ¬ ¬ 𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  STAB wstab 816  DECID wdc 820 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 604  ax-in2 605  ax-io 699 This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821 This theorem is referenced by: (None)
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