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Mirrors > Home > ILE Home > Th. List > eqer | Unicode version |
Description: Equivalence relation involving equality of dependent classes and . (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
eqer.1 | |
eqer.2 |
Ref | Expression |
---|---|
eqer |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqer.2 | . . . . 5 | |
2 | 1 | relopabi 4463 | . . . 4 |
3 | 2 | a1i 9 | . . 3 |
4 | id 19 | . . . . . 6 | |
5 | 4 | eqcomd 2045 | . . . . 5 |
6 | eqer.1 | . . . . . 6 | |
7 | 6, 1 | eqerlem 6137 | . . . . 5 |
8 | 6, 1 | eqerlem 6137 | . . . . 5 |
9 | 5, 7, 8 | 3imtr4i 190 | . . . 4 |
10 | 9 | adantl 262 | . . 3 |
11 | eqtr 2057 | . . . . 5 | |
12 | 6, 1 | eqerlem 6137 | . . . . . 6 |
13 | 7, 12 | anbi12i 433 | . . . . 5 |
14 | 6, 1 | eqerlem 6137 | . . . . 5 |
15 | 11, 13, 14 | 3imtr4i 190 | . . . 4 |
16 | 15 | adantl 262 | . . 3 |
17 | vex 2560 | . . . . 5 | |
18 | eqid 2040 | . . . . . 6 | |
19 | 6, 1 | eqerlem 6137 | . . . . . 6 |
20 | 18, 19 | mpbir 134 | . . . . 5 |
21 | 17, 20 | 2th 163 | . . . 4 |
22 | 21 | a1i 9 | . . 3 |
23 | 3, 10, 16, 22 | iserd 6132 | . 2 |
24 | 23 | trud 1252 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wtru 1244 wcel 1393 cvv 2557 csb 2852 class class class wbr 3764 copab 3817 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-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-er 6106 |
This theorem is referenced by: ider 6139 |
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