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Mirrors > Home > MPE Home > Th. List > Mathboxes > sbcel12gOLD | Structured version Visualization version Unicode version |
Description: Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) Obsolete as of 18-Aug-2018. Use sbcel12 3772 instead. (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
sbcel12gOLD |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq2 3270 |
. . 3
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2 | dfsbcq2 3270 |
. . . . 5
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3 | 2 | abbidv 2569 |
. . . 4
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4 | dfsbcq2 3270 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 4 | abbidv 2569 |
. . . 4
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6 | 3, 5 | eleq12d 2523 |
. . 3
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7 | nfs1v 2266 |
. . . . . 6
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8 | 7 | nfab 2596 |
. . . . 5
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9 | nfs1v 2266 |
. . . . . 6
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10 | 9 | nfab 2596 |
. . . . 5
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11 | 8, 10 | nfel 2604 |
. . . 4
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12 | sbab 2578 |
. . . . 5
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13 | sbab 2578 |
. . . . 5
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14 | 12, 13 | eleq12d 2523 |
. . . 4
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15 | 11, 14 | sbie 2237 |
. . 3
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16 | 1, 6, 15 | vtoclbg 3108 |
. 2
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17 | df-csb 3364 |
. . 3
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18 | df-csb 3364 |
. . 3
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19 | 17, 18 | eleq12i 2522 |
. 2
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20 | 16, 19 | syl6bbr 267 |
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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-v 3047 df-sbc 3268 df-csb 3364 |
This theorem is referenced by: sbcel2gOLD 36906 csbxpgVD 37291 csbrngVD 37293 |
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