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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq2 2916 | . . 3 | |
2 | dfsbcq2 2916 | . . . . 5 | |
3 | 2 | abbidv 2258 | . . . 4 |
4 | dfsbcq2 2916 | . . . . 5 | |
5 | 4 | abbidv 2258 | . . . 4 |
6 | 3, 5 | eleq12d 2211 | . . 3 |
7 | nfs1v 1913 | . . . . . 6 | |
8 | 7 | nfab 2287 | . . . . 5 |
9 | nfs1v 1913 | . . . . . 6 | |
10 | 9 | nfab 2287 | . . . . 5 |
11 | 8, 10 | nfel 2291 | . . . 4 |
12 | sbab 2268 | . . . . 5 | |
13 | sbab 2268 | . . . . 5 | |
14 | 12, 13 | eleq12d 2211 | . . . 4 |
15 | 11, 14 | sbie 1765 | . . 3 |
16 | 1, 6, 15 | vtoclbg 2750 | . 2 |
17 | df-csb 3008 | . . 3 | |
18 | df-csb 3008 | . . 3 | |
19 | 17, 18 | eleq12i 2208 | . 2 |
20 | 16, 19 | syl6bbr 197 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1332 wcel 1481 wsb 1736 cab 2126 wsbc 2913 csb 3007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-sbc 2914 df-csb 3008 |
This theorem is referenced by: sbcnel12g 3024 sbcel1g 3026 sbcel2g 3028 sbccsb2g 3037 ixpsnval 6603 |
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