Theorem dcnn 843

Description: Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 837. The relation between dcn 837 and dcnn 843 is analogous to that between notnot 624 and notnotnot 629 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 843 means that a proposition is testable if and only if its negation is testable, and dcn 837 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 837	. 2 ⊢ (DECID ¬ 𝜑 → DECID ¬ ¬ 𝜑)
2		stabnot 828	. . 3 ⊢ STAB ¬ 𝜑
3		stdcn 842	. . 3 ⊢ (STAB ¬ 𝜑 ↔ (DECID ¬ ¬ 𝜑 → DECID ¬ 𝜑))
4	2, 3	mpbi 144	. 2 ⊢ (DECID ¬ ¬ 𝜑 → DECID ¬ 𝜑)
5	1, 4	impbii 125	1 ⊢ (DECID ¬ 𝜑 ↔ DECID ¬ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  STAB wstab 825  DECID wdc 829

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 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830

This theorem is referenced by: (None)