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Mirrors > Home > ILE Home > Th. List > dcim | Unicode version |
Description: An implication between two decidable propositions is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.) |
Ref | Expression |
---|---|
dcim | DECID DECID DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 825 | . 2 DECID | |
2 | df-dc 825 | . . . . . . . 8 DECID | |
3 | 2 | anbi2i 453 | . . . . . . 7 DECID |
4 | andi 808 | . . . . . . 7 | |
5 | 3, 4 | bitri 183 | . . . . . 6 DECID |
6 | pm3.4 331 | . . . . . . 7 | |
7 | annimim 676 | . . . . . . 7 | |
8 | 6, 7 | orim12i 749 | . . . . . 6 |
9 | 5, 8 | sylbi 120 | . . . . 5 DECID |
10 | df-dc 825 | . . . . 5 DECID | |
11 | 9, 10 | sylibr 133 | . . . 4 DECID DECID |
12 | 11 | ex 114 | . . 3 DECID DECID |
13 | ax-in2 605 | . . . . 5 | |
14 | 13 | a1d 22 | . . . 4 DECID |
15 | orc 702 | . . . . 5 | |
16 | 15, 10 | sylibr 133 | . . . 4 DECID |
17 | 14, 16 | syl6 33 | . . 3 DECID DECID |
18 | 12, 17 | jaoi 706 | . 2 DECID DECID |
19 | 1, 18 | sylbi 120 | 1 DECID DECID DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 698 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-dc 825 |
This theorem is referenced by: pm4.79dc 893 pm5.11dc 899 dcbi 925 annimdc 926 |
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