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Theorem dcnn 816
Description: Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 810. The relation between dcn 810 and dcnn 816 is analogous to that between notnot 601 and notnotnot 606 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 816 means that a proposition is testable if and only if its negation is testable, and dcn 810 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 810 . 2 (DECID ¬ 𝜑DECID ¬ ¬ 𝜑)
2 stabnot 801 . . 3 STAB ¬ 𝜑
3 stdcn 815 . . 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 798  DECID wdc 802
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 586  ax-in2 587  ax-io 681
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803
This theorem is referenced by: (None)
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