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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 2958 | . . 3 | |
2 | dfsbcq2 2958 | . . . . 5 | |
3 | 2 | abbidv 2288 | . . . 4 |
4 | dfsbcq2 2958 | . . . . 5 | |
5 | 4 | abbidv 2288 | . . . 4 |
6 | 3, 5 | eqeq12d 2185 | . . 3 |
7 | nfs1v 1932 | . . . . . 6 | |
8 | 7 | nfab 2317 | . . . . 5 |
9 | nfs1v 1932 | . . . . . 6 | |
10 | 9 | nfab 2317 | . . . . 5 |
11 | 8, 10 | nfeq 2320 | . . . 4 |
12 | sbab 2298 | . . . . 5 | |
13 | sbab 2298 | . . . . 5 | |
14 | 12, 13 | eqeq12d 2185 | . . . 4 |
15 | 11, 14 | sbie 1784 | . . 3 |
16 | 1, 6, 15 | vtoclbg 2791 | . 2 |
17 | df-csb 3050 | . . 3 | |
18 | df-csb 3050 | . . 3 | |
19 | 17, 18 | eqeq12i 2184 | . 2 |
20 | 16, 19 | bitr4di 197 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1348 wsb 1755 wcel 2141 cab 2156 wsbc 2955 csb 3049 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-sbc 2956 df-csb 3050 |
This theorem is referenced by: sbcne12g 3067 sbceq1g 3069 sbceq2g 3071 sbcfng 5345 fprodmodd 11604 |
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