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Mirrors > Home > ILE Home > Th. List > ereq1 | Unicode version |
Description: Equality theorem for equivalence predicate. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
ereq1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | releq 4422 | . . 3 | |
2 | dmeq 4535 | . . . 4 | |
3 | 2 | eqeq1d 2048 | . . 3 |
4 | cnveq 4509 | . . . . . 6 | |
5 | coeq1 4493 | . . . . . . 7 | |
6 | coeq2 4494 | . . . . . . 7 | |
7 | 5, 6 | eqtrd 2072 | . . . . . 6 |
8 | 4, 7 | uneq12d 3098 | . . . . 5 |
9 | 8 | sseq1d 2972 | . . . 4 |
10 | sseq2 2967 | . . . 4 | |
11 | 9, 10 | bitrd 177 | . . 3 |
12 | 1, 3, 11 | 3anbi123d 1207 | . 2 |
13 | df-er 6106 | . 2 | |
14 | df-er 6106 | . 2 | |
15 | 12, 13, 14 | 3bitr4g 212 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 98 w3a 885 wceq 1243 cun 2915 wss 2917 ccnv 4344 cdm 4345 ccom 4349 wrel 4350 wer 6103 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-er 6106 |
This theorem is referenced by: riinerm 6179 |
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