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Mirrors > Home > MPE Home > Th. List > sbcel12 | Structured 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 3297 |
. . . 4
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2 | dfsbcq2 3297 |
. . . . . 6
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3 | 2 | abbidv 2590 |
. . . . 5
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4 | dfsbcq2 3297 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 4 | abbidv 2590 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | 3, 5 | eleq12d 2536 |
. . . 4
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7 | nfs1v 2151 |
. . . . . . 7
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8 | 7 | nfab 2620 |
. . . . . 6
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9 | nfs1v 2151 |
. . . . . . 7
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10 | 9 | nfab 2620 |
. . . . . 6
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11 | 8, 10 | nfel 2629 |
. . . . 5
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12 | sbab 2601 |
. . . . . 6
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13 | sbab 2601 |
. . . . . 6
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14 | 12, 13 | eleq12d 2536 |
. . . . 5
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15 | 11, 14 | sbie 2110 |
. . . 4
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16 | 1, 6, 15 | vtoclbg 3137 |
. . 3
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17 | df-csb 3399 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | df-csb 3399 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 17, 18 | eleq12i 2533 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 16, 19 | syl6bbr 263 |
. 2
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21 | sbcex 3304 |
. . . 4
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22 | 21 | con3i 135 |
. . 3
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23 | noel 3752 |
. . . 4
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24 | csbprc 3784 |
. . . . . 6
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25 | 24 | eleq2d 2524 |
. . . . 5
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26 | 25 | notbid 294 |
. . . 4
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27 | 23, 26 | mpbiri 233 |
. . 3
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28 | 22, 27 | 2falsed 351 |
. 2
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29 | 20, 28 | pm2.61i 164 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-tru 1373 df-fal 1376 df-ex 1588 df-nf 1591 df-sb 1703 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-v 3080 df-sbc 3295 df-csb 3399 df-dif 3442 df-in 3446 df-ss 3453 df-nul 3749 |
This theorem is referenced by: sbcnel12g 3789 sbcel1g 3792 sbcel2 3794 sbccsb2 3814 csbmpt12 4733 ixpsnval 7379 fmptdF 26150 finixpnum 28585 |
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