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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.) |
Ref | Expression |
---|---|
dcnn | DECID DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dcn 837 | . 2 DECID DECID | |
2 | stabnot 828 | . . 3 STAB | |
3 | stdcn 842 | . . 3 STAB DECID DECID | |
4 | 2, 3 | mpbi 144 | . 2 DECID DECID |
5 | 1, 4 | impbii 125 | 1 DECID DECID |
Colors of variables: wff set class |
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) |
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