Theorem dcnn 853

Description: Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 847. The relation between dcn 847 and dcnn 853 is analogous to that between notnot 632 and notnotnot 637 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 853 means that a proposition is testable if and only if its negation is testable, and dcn 847 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 847	. 2 ⊢ (DECID ¬ 𝜑 → DECID ¬ ¬ 𝜑)
2		stabnot 838	. . 3 ⊢ STAB ¬ 𝜑
3		stdcn 852	. . 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 835  DECID wdc 839

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 617  ax-in2 618  ax-io 714

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840

This theorem is referenced by: (None)