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Mirrors > Home > ILE Home > Th. List > sbcel12g | Unicode version |
Description: Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
sbcel12g |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq2 2980 |
. . 3
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2 | dfsbcq2 2980 |
. . . . 5
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3 | 2 | abbidv 2307 |
. . . 4
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4 | dfsbcq2 2980 |
. . . . 5
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5 | 4 | abbidv 2307 |
. . . 4
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6 | 3, 5 | eleq12d 2260 |
. . 3
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7 | nfs1v 1951 |
. . . . . 6
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8 | 7 | nfab 2337 |
. . . . 5
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9 | nfs1v 1951 |
. . . . . 6
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10 | 9 | nfab 2337 |
. . . . 5
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11 | 8, 10 | nfel 2341 |
. . . 4
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12 | sbab 2317 |
. . . . 5
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13 | sbab 2317 |
. . . . 5
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14 | 12, 13 | eleq12d 2260 |
. . . 4
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15 | 11, 14 | sbie 1802 |
. . 3
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16 | 1, 6, 15 | vtoclbg 2813 |
. 2
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17 | df-csb 3073 |
. . 3
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18 | df-csb 3073 |
. . 3
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19 | 17, 18 | eleq12i 2257 |
. 2
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20 | 16, 19 | bitr4di 198 |
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-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-sbc 2978 df-csb 3073 |
This theorem is referenced by: sbcnel12g 3089 sbcel1g 3091 sbcel2g 3093 sbccsb2g 3102 ixpsnval 6728 |
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