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Mirrors > Home > ILE Home > Th. List > ecovcom | Unicode version |
Description: Lemma used to transfer a commutative law via an equivalence relation. Most uses will want ecovicom 6545 instead. (Contributed by NM, 29-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.) |
Ref | Expression |
---|---|
ecovcom.1 | |
ecovcom.2 | |
ecovcom.3 | |
ecovcom.4 | |
ecovcom.5 |
Ref | Expression |
---|---|
ecovcom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecovcom.1 | . 2 | |
2 | oveq1 5789 | . . 3 | |
3 | oveq2 5790 | . . 3 | |
4 | 2, 3 | eqeq12d 2155 | . 2 |
5 | oveq2 5790 | . . 3 | |
6 | oveq1 5789 | . . 3 | |
7 | 5, 6 | eqeq12d 2155 | . 2 |
8 | ecovcom.4 | . . . 4 | |
9 | ecovcom.5 | . . . 4 | |
10 | opeq12 3715 | . . . . 5 | |
11 | 10 | eceq1d 6473 | . . . 4 |
12 | 8, 9, 11 | mp2an 423 | . . 3 |
13 | ecovcom.2 | . . 3 | |
14 | ecovcom.3 | . . . 4 | |
15 | 14 | ancoms 266 | . . 3 |
16 | 12, 13, 15 | 3eqtr4a 2199 | . 2 |
17 | 1, 4, 7, 16 | 2ecoptocl 6525 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1332 wcel 1481 cop 3535 cxp 4545 (class class class)co 5782 cec 6435 cqs 6436 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-cnv 4555 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fv 5139 df-ov 5785 df-ec 6439 df-qs 6443 |
This theorem is referenced by: (None) |
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