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Mirrors > Home > ILE Home > Th. List > stdcndc | Unicode version |
Description: A formula is decidable if and only if its negation is decidable and it is stable (that is, it is testable and stable). (Contributed by David A. Wheeler, 13-Aug-2018.) (Proof shortened by BJ, 28-Oct-2023.) |
Ref | Expression |
---|---|
stdcndc | STAB DECID DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-stab 817 | . . . 4 STAB | |
2 | df-dc 821 | . . . 4 DECID | |
3 | pm2.36 794 | . . . . 5 | |
4 | 3 | imp 123 | . . . 4 |
5 | 1, 2, 4 | syl2anb 289 | . . 3 STAB DECID |
6 | df-dc 821 | . . 3 DECID | |
7 | 5, 6 | sylibr 133 | . 2 STAB DECID DECID |
8 | dcstab 830 | . . 3 DECID STAB | |
9 | dcn 828 | . . 3 DECID DECID | |
10 | 8, 9 | jca 304 | . 2 DECID STAB DECID |
11 | 7, 10 | impbii 125 | 1 STAB DECID DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 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: stdcn 833 |
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