Theorem dcnn 838

Description: Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 832. The relation between dcn 832 and dcnn 838 is analogous to that between notnot 619 and notnotnot 624 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 838 means that a proposition is testable if and only if its negation is testable, and dcn 832 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 832	. 2 ⊢ (DECID ¬ 𝜑 → DECID ¬ ¬ 𝜑)
2		stabnot 823	. . 3 ⊢ STAB ¬ 𝜑
3		stdcn 837	. . 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 820  DECID wdc 824

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 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825

This theorem is referenced by: (None)