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Mirrors > Home > MPE Home > Th. List > sbcel12 | Structured version Visualization version Unicode version |
Description: Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Revised by NM, 18-Aug-2018.) |
Ref | Expression |
---|---|
sbcel12 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq2 3238 |
. . . 4
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2 | dfsbcq2 3238 |
. . . . . 6
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3 | 2 | abbidv 2570 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | dfsbcq2 3238 |
. . . . . 6
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5 | 4 | abbidv 2570 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | 3, 5 | eleq12d 2524 |
. . . 4
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7 | nfs1v 2267 |
. . . . . . 7
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8 | 7 | nfab 2597 |
. . . . . 6
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9 | nfs1v 2267 |
. . . . . . 7
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10 | 9 | nfab 2597 |
. . . . . 6
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11 | 8, 10 | nfel 2605 |
. . . . 5
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12 | sbab 2579 |
. . . . . 6
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13 | sbab 2579 |
. . . . . 6
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14 | 12, 13 | eleq12d 2524 |
. . . . 5
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15 | 11, 14 | sbie 2238 |
. . . 4
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16 | 1, 6, 15 | vtoclbg 3076 |
. . 3
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17 | df-csb 3332 |
. . . 4
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18 | df-csb 3332 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 17, 18 | eleq12i 2523 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 16, 19 | syl6bbr 271 |
. 2
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21 | sbcex 3245 |
. . . 4
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22 | 21 | con3i 142 |
. . 3
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23 | noel 3703 |
. . . 4
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24 | csbprc 3738 |
. . . . 5
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25 | 24 | eleq2d 2515 |
. . . 4
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26 | 23, 25 | mtbiri 309 |
. . 3
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27 | 22, 26 | 2falsed 357 |
. 2
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28 | 20, 27 | pm2.61i 169 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1673 ax-4 1686 ax-5 1762 ax-6 1809 ax-7 1855 ax-10 1919 ax-11 1924 ax-12 1937 ax-13 2092 ax-ext 2432 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-tru 1451 df-fal 1454 df-ex 1668 df-nf 1672 df-sb 1802 df-clab 2439 df-cleq 2445 df-clel 2448 df-nfc 2582 df-v 3015 df-sbc 3236 df-csb 3332 df-dif 3375 df-in 3379 df-ss 3386 df-nul 3700 |
This theorem is referenced by: sbcnel12g 3742 sbcel1g 3744 sbcel2 3746 sbccsb2 3762 csbmpt12 4708 ixpsnval 7512 fmptdF 28264 csbmpt22g 31734 csbfinxpg 31782 finixpnum 31932 |
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