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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 6537 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 5781 | . . 3 | |
3 | oveq2 5782 | . . 3 | |
4 | 2, 3 | eqeq12d 2154 | . 2 |
5 | oveq2 5782 | . . 3 | |
6 | oveq1 5781 | . . 3 | |
7 | 5, 6 | eqeq12d 2154 | . 2 |
8 | ecovcom.4 | . . . 4 | |
9 | ecovcom.5 | . . . 4 | |
10 | opeq12 3707 | . . . . 5 | |
11 | 10 | eceq1d 6465 | . . . 4 |
12 | 8, 9, 11 | mp2an 422 | . . 3 |
13 | ecovcom.2 | . . 3 | |
14 | ecovcom.3 | . . . 4 | |
15 | 14 | ancoms 266 | . . 3 |
16 | 12, 13, 15 | 3eqtr4a 2198 | . 2 |
17 | 1, 4, 7, 16 | 2ecoptocl 6517 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cop 3530 cxp 4537 (class class class)co 5774 cec 6427 cqs 6428 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fv 5131 df-ov 5777 df-ec 6431 df-qs 6435 |
This theorem is referenced by: (None) |
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