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Description: Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 828. The relation between dcn 828 and dcnn 834 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 834 means that a proposition is testable if and only if its negation is testable, and dcn 828 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 828 | . 2 DECID DECID | |
2 | stabnot 819 | . . 3 STAB | |
3 | stdcn 833 | . . 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 816 DECID wdc 820 |
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 817 df-dc 821 |
This theorem is referenced by: (None) |
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