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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 3995 |
. . . . . . 7
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2 | 1 | bj-snex 12692 |
. . . . . 6
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3 | 2 | zfauscl 3988 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | eleq1 2162 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | eleq1 2162 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 5 | anbi1d 456 |
. . . . . . 7
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7 | 4, 6 | bibi12d 234 |
. . . . . 6
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8 | 1, 7 | spcv 2734 |
. . . . 5
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9 | 3, 8 | eximii 1549 |
. . . 4
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10 | 1 | snid 3503 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() |
11 | 10 | biantrur 299 |
. . . . . . 7
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12 | 11 | bicomi 131 |
. . . . . 6
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13 | 12 | bibi2i 226 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 13 | exbii 1552 |
. . . 4
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15 | 9, 14 | mpbi 144 |
. . 3
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16 | bj-bd0el 12647 |
. . . . 5
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17 | 16 | ax-bj-d0cl 12703 |
. . . 4
![]() ![]() ![]() ![]() |
18 | bj-dcbi 12707 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 17, 18 | mpbii 147 |
. . 3
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20 | 15, 19 | eximii 1549 |
. 2
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21 | bj-ex 12551 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
22 | 20, 21 | ax-mp 7 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-nul 3994 ax-pr 4069 ax-bd0 12592 ax-bdim 12593 ax-bdor 12595 ax-bdn 12596 ax-bdal 12597 ax-bdex 12598 ax-bdeq 12599 ax-bdsep 12663 ax-bj-d0cl 12703 |
This theorem depends on definitions: df-bi 116 df-dc 787 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-sn 3480 df-pr 3481 df-bdc 12620 |
This theorem is referenced by: (None) |
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