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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 2940 | . . 3 | |
2 | dfsbcq2 2940 | . . . . 5 | |
3 | 2 | abbidv 2275 | . . . 4 |
4 | dfsbcq2 2940 | . . . . 5 | |
5 | 4 | abbidv 2275 | . . . 4 |
6 | 3, 5 | eqeq12d 2172 | . . 3 |
7 | nfs1v 1919 | . . . . . 6 | |
8 | 7 | nfab 2304 | . . . . 5 |
9 | nfs1v 1919 | . . . . . 6 | |
10 | 9 | nfab 2304 | . . . . 5 |
11 | 8, 10 | nfeq 2307 | . . . 4 |
12 | sbab 2285 | . . . . 5 | |
13 | sbab 2285 | . . . . 5 | |
14 | 12, 13 | eqeq12d 2172 | . . . 4 |
15 | 11, 14 | sbie 1771 | . . 3 |
16 | 1, 6, 15 | vtoclbg 2773 | . 2 |
17 | df-csb 3032 | . . 3 | |
18 | df-csb 3032 | . . 3 | |
19 | 17, 18 | eqeq12i 2171 | . 2 |
20 | 16, 19 | bitr4di 197 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1335 wsb 1742 wcel 2128 cab 2143 wsbc 2937 csb 3031 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-sbc 2938 df-csb 3032 |
This theorem is referenced by: sbcne12g 3049 sbceq1g 3051 sbceq2g 3053 sbcfng 5317 fprodmodd 11538 |
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