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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-d0clsepcl | Unicode version |
Description: Δ0-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-d0clsepcl | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4103 | . . . . . . 7 | |
2 | 1 | bj-snex 13630 | . . . . . 6 |
3 | 2 | zfauscl 4096 | . . . . 5 |
4 | eleq1 2227 | . . . . . . 7 | |
5 | eleq1 2227 | . . . . . . . 8 | |
6 | 5 | anbi1d 461 | . . . . . . 7 |
7 | 4, 6 | bibi12d 234 | . . . . . 6 |
8 | 1, 7 | spcv 2815 | . . . . 5 |
9 | 3, 8 | eximii 1589 | . . . 4 |
10 | 1 | snid 3601 | . . . . . . . 8 |
11 | 10 | biantrur 301 | . . . . . . 7 |
12 | 11 | bicomi 131 | . . . . . 6 |
13 | 12 | bibi2i 226 | . . . . 5 |
14 | 13 | exbii 1592 | . . . 4 |
15 | 9, 14 | mpbi 144 | . . 3 |
16 | bj-bd0el 13585 | . . . . 5 BOUNDED | |
17 | 16 | ax-bj-d0cl 13641 | . . . 4 DECID |
18 | dcbiit 829 | . . . 4 DECID DECID | |
19 | 17, 18 | mpbii 147 | . . 3 DECID |
20 | 15, 19 | eximii 1589 | . 2 DECID |
21 | bj-ex 13478 | . 2 DECID DECID | |
22 | 20, 21 | ax-mp 5 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 DECID wdc 824 wal 1340 wceq 1342 wex 1479 wcel 2135 c0 3404 csn 3570 |
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 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pr 4181 ax-bd0 13530 ax-bdim 13531 ax-bdor 13533 ax-bdn 13534 ax-bdal 13535 ax-bdex 13536 ax-bdeq 13537 ax-bdsep 13601 ax-bj-d0cl 13641 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-sn 3576 df-pr 3577 df-bdc 13558 |
This theorem is referenced by: (None) |
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