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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 2885 | . . 3 | |
2 | dfsbcq2 2885 | . . . . 5 | |
3 | 2 | abbidv 2235 | . . . 4 |
4 | dfsbcq2 2885 | . . . . 5 | |
5 | 4 | abbidv 2235 | . . . 4 |
6 | 3, 5 | eleq12d 2188 | . . 3 |
7 | nfs1v 1892 | . . . . . 6 | |
8 | 7 | nfab 2263 | . . . . 5 |
9 | nfs1v 1892 | . . . . . 6 | |
10 | 9 | nfab 2263 | . . . . 5 |
11 | 8, 10 | nfel 2267 | . . . 4 |
12 | sbab 2244 | . . . . 5 | |
13 | sbab 2244 | . . . . 5 | |
14 | 12, 13 | eleq12d 2188 | . . . 4 |
15 | 11, 14 | sbie 1749 | . . 3 |
16 | 1, 6, 15 | vtoclbg 2721 | . 2 |
17 | df-csb 2976 | . . 3 | |
18 | df-csb 2976 | . . 3 | |
19 | 17, 18 | eleq12i 2185 | . 2 |
20 | 16, 19 | syl6bbr 197 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1316 wcel 1465 wsb 1720 cab 2103 wsbc 2882 csb 2975 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-sbc 2883 df-csb 2976 |
This theorem is referenced by: sbcnel12g 2990 sbcel1g 2992 sbcel2g 2994 sbccsb2g 3002 ixpsnval 6563 |
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