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Mirrors > Home > ILE Home > Th. List > ecovicom | Unicode version |
Description: Lemma used to transfer a commutative law via an equivalence relation. (Contributed by Jim Kingdon, 15-Sep-2019.) |
Ref | Expression |
---|---|
ecovicom.1 | |
ecovicom.2 | |
ecovicom.3 | |
ecovicom.4 | |
ecovicom.5 |
Ref | Expression |
---|---|
ecovicom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecovicom.1 | . 2 | |
2 | oveq1 5519 | . . 3 | |
3 | oveq2 5520 | . . 3 | |
4 | 2, 3 | eqeq12d 2054 | . 2 |
5 | oveq2 5520 | . . 3 | |
6 | oveq1 5519 | . . 3 | |
7 | 5, 6 | eqeq12d 2054 | . 2 |
8 | ecovicom.4 | . . . 4 | |
9 | ecovicom.5 | . . . 4 | |
10 | opeq12 3551 | . . . . 5 | |
11 | 10 | eceq1d 6142 | . . . 4 |
12 | 8, 9, 11 | syl2anc 391 | . . 3 |
13 | ecovicom.2 | . . 3 | |
14 | ecovicom.3 | . . . 4 | |
15 | 14 | ancoms 255 | . . 3 |
16 | 12, 13, 15 | 3eqtr4d 2082 | . 2 |
17 | 1, 4, 7, 16 | 2ecoptocl 6194 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wcel 1393 cop 3378 cxp 4343 (class class class)co 5512 cec 6104 cqs 6105 |
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-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-cnv 4353 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fv 4910 df-ov 5515 df-ec 6108 df-qs 6112 |
This theorem is referenced by: addcomnqg 6479 mulcomnqg 6481 addcomsrg 6840 mulcomsrg 6842 axmulcom 6945 |
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