Theorem dcnn 849

Description: Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 843. The relation between dcn 843 and dcnn 849 is analogous to that between notnot 630 and notnotnot 635 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 849 means that a proposition is testable if and only if its negation is testable, and dcn 843 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 ¬ 𝜑 ↔ DECID ¬ ¬ 𝜑)

Proof of Theorem dcnn

Step	Hyp	Ref	Expression
1		dcn 843	. 2 ⊢ (DECID ¬ 𝜑 → DECID ¬ ¬ 𝜑)
2		stabnot 834	. . 3 ⊢ STAB ¬ 𝜑
3		stdcn 848	. . 3 ⊢ (STAB ¬ 𝜑 ↔ (DECID ¬ ¬ 𝜑 → DECID ¬ 𝜑))
4	2, 3	mpbi 145	. 2 ⊢ (DECID ¬ ¬ 𝜑 → DECID ¬ 𝜑)
5	1, 4	impbii 126	1 ⊢ (DECID ¬ 𝜑 ↔ DECID ¬ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105  STAB wstab 831  DECID wdc 835

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 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836

This theorem is referenced by: (None)