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Mirrors > Home > ILE Home > Th. List > ifmdc | Unicode version |
Description: If a conditional class is inhabited, then the condition is decidable. This shows that conditionals are not very useful unless one can prove the condition decidable. (Contributed by BJ, 24-Sep-2022.) |
Ref | Expression |
---|---|
ifmdc | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2202 | . . . . 5 | |
2 | 1 | imbi1d 230 | . . . 4 |
3 | df-if 3475 | . . . . . 6 | |
4 | 3 | abeq2i 2250 | . . . . 5 |
5 | simpr 109 | . . . . . 6 | |
6 | simpr 109 | . . . . . 6 | |
7 | 5, 6 | orim12i 748 | . . . . 5 |
8 | 4, 7 | sylbi 120 | . . . 4 |
9 | 2, 8 | vtoclg 2746 | . . 3 |
10 | 9 | pm2.43i 49 | . 2 |
11 | df-dc 820 | . 2 DECID | |
12 | 10, 11 | sylibr 133 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 DECID wdc 819 wceq 1331 wcel 1480 cif 3474 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-if 3475 |
This theorem is referenced by: (None) |
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