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Mirrors > Home > ILE Home > Th. List > sbceqg | Unicode version |
Description: Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
sbceqg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq2 2818 | . . 3 | |
2 | dfsbcq2 2818 | . . . . 5 | |
3 | 2 | abbidv 2196 | . . . 4 |
4 | dfsbcq2 2818 | . . . . 5 | |
5 | 4 | abbidv 2196 | . . . 4 |
6 | 3, 5 | eqeq12d 2095 | . . 3 |
7 | nfs1v 1856 | . . . . . 6 | |
8 | 7 | nfab 2223 | . . . . 5 |
9 | nfs1v 1856 | . . . . . 6 | |
10 | 9 | nfab 2223 | . . . . 5 |
11 | 8, 10 | nfeq 2226 | . . . 4 |
12 | sbab 2205 | . . . . 5 | |
13 | sbab 2205 | . . . . 5 | |
14 | 12, 13 | eqeq12d 2095 | . . . 4 |
15 | 11, 14 | sbie 1714 | . . 3 |
16 | 1, 6, 15 | vtoclbg 2659 | . 2 |
17 | df-csb 2909 | . . 3 | |
18 | df-csb 2909 | . . 3 | |
19 | 17, 18 | eqeq12i 2094 | . 2 |
20 | 16, 19 | syl6bbr 196 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 103 wceq 1284 wcel 1433 wsb 1685 cab 2067 wsbc 2815 csb 2908 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-sbc 2816 df-csb 2909 |
This theorem is referenced by: sbcne12g 2924 sbceq1g 2926 sbceq2g 2928 sbcfng 5064 |
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