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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 4724 | . . . 4 |
3 | 2 | a1i 9 | . . 3 |
4 | id 19 | . . . . . 6 | |
5 | 4 | eqcomd 2170 | . . . . 5 |
6 | eqer.1 | . . . . . 6 | |
7 | 6, 1 | eqerlem 6523 | . . . . 5 |
8 | 6, 1 | eqerlem 6523 | . . . . 5 |
9 | 5, 7, 8 | 3imtr4i 200 | . . . 4 |
10 | 9 | adantl 275 | . . 3 |
11 | eqtr 2182 | . . . . 5 | |
12 | 6, 1 | eqerlem 6523 | . . . . . 6 |
13 | 7, 12 | anbi12i 456 | . . . . 5 |
14 | 6, 1 | eqerlem 6523 | . . . . 5 |
15 | 11, 13, 14 | 3imtr4i 200 | . . . 4 |
16 | 15 | adantl 275 | . . 3 |
17 | vex 2724 | . . . . 5 | |
18 | eqid 2164 | . . . . . 6 | |
19 | 6, 1 | eqerlem 6523 | . . . . . 6 |
20 | 18, 19 | mpbir 145 | . . . . 5 |
21 | 17, 20 | 2th 173 | . . . 4 |
22 | 21 | a1i 9 | . . 3 |
23 | 3, 10, 16, 22 | iserd 6518 | . 2 |
24 | 23 | mptru 1351 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wtru 1343 wcel 2135 cvv 2721 csb 3040 class class class wbr 3976 copab 4036 wrel 4603 wer 6489 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-csb 3041 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-er 6492 |
This theorem is referenced by: ider 6525 |
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