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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 4481 | . . . 4 |
3 | 2 | a1i 9 | . . 3 |
4 | id 19 | . . . . . 6 | |
5 | 4 | eqcomd 2086 | . . . . 5 |
6 | eqer.1 | . . . . . 6 | |
7 | 6, 1 | eqerlem 6160 | . . . . 5 |
8 | 6, 1 | eqerlem 6160 | . . . . 5 |
9 | 5, 7, 8 | 3imtr4i 199 | . . . 4 |
10 | 9 | adantl 271 | . . 3 |
11 | eqtr 2098 | . . . . 5 | |
12 | 6, 1 | eqerlem 6160 | . . . . . 6 |
13 | 7, 12 | anbi12i 447 | . . . . 5 |
14 | 6, 1 | eqerlem 6160 | . . . . 5 |
15 | 11, 13, 14 | 3imtr4i 199 | . . . 4 |
16 | 15 | adantl 271 | . . 3 |
17 | vex 2604 | . . . . 5 | |
18 | eqid 2081 | . . . . . 6 | |
19 | 6, 1 | eqerlem 6160 | . . . . . 6 |
20 | 18, 19 | mpbir 144 | . . . . 5 |
21 | 17, 20 | 2th 172 | . . . 4 |
22 | 21 | a1i 9 | . . 3 |
23 | 3, 10, 16, 22 | iserd 6155 | . 2 |
24 | 23 | trud 1293 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wtru 1285 wcel 1433 cvv 2601 csb 2908 class class class wbr 3785 copab 3838 wrel 4368 wer 6126 |
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-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-er 6129 |
This theorem is referenced by: ider 6162 |
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