Theorem dcnn 856

Description: Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 850. The relation between dcn 850 and dcnn 856 is analogous to that between notnot 634 and notnotnot 639 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 856 means that a proposition is testable if and only if its negation is testable, and dcn 850 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 850	. 2 ⊢ (DECID ¬ 𝜑 → DECID ¬ ¬ 𝜑)
2		stabnot 841	. . 3 ⊢ STAB ¬ 𝜑
3		stdcn 855	. . 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 838  DECID wdc 842

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 619  ax-in2 620  ax-io 717

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843

This theorem is referenced by: (None)