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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 4142 |
. . . . . . 7
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2 | 1 | bj-snex 14936 |
. . . . . 6
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3 | 2 | zfauscl 4135 |
. . . . 5
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4 | eleq1 2250 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | eleq1 2250 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 5 | anbi1d 465 |
. . . . . . 7
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7 | 4, 6 | bibi12d 235 |
. . . . . 6
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8 | 1, 7 | spcv 2843 |
. . . . 5
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9 | 3, 8 | eximii 1612 |
. . . 4
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10 | 1 | snid 3635 |
. . . . . . . 8
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11 | 10 | biantrur 303 |
. . . . . . 7
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12 | 11 | bicomi 132 |
. . . . . 6
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13 | 12 | bibi2i 227 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 13 | exbii 1615 |
. . . 4
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15 | 9, 14 | mpbi 145 |
. . 3
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16 | bj-bd0el 14891 |
. . . . 5
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17 | 16 | ax-bj-d0cl 14947 |
. . . 4
![]() ![]() ![]() ![]() |
18 | dcbiit 840 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 17, 18 | mpbii 148 |
. . 3
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20 | 15, 19 | eximii 1612 |
. 2
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21 | bj-ex 14785 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
22 | 20, 21 | ax-mp 5 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-nul 4141 ax-pr 4221 ax-bd0 14836 ax-bdim 14837 ax-bdor 14839 ax-bdn 14840 ax-bdal 14841 ax-bdex 14842 ax-bdeq 14843 ax-bdsep 14907 ax-bj-d0cl 14947 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-sn 3610 df-pr 3611 df-bdc 14864 |
This theorem is referenced by: (None) |
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