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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 6668 instead. (Contributed by NM, 29-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.) |
Ref | Expression |
---|---|
ecovcom.1 |
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ecovcom.2 |
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ecovcom.3 |
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ecovcom.4 |
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ecovcom.5 |
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Ref | Expression |
---|---|
ecovcom |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecovcom.1 |
. 2
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2 | oveq1 5902 |
. . 3
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3 | oveq2 5903 |
. . 3
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4 | 2, 3 | eqeq12d 2204 |
. 2
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5 | oveq2 5903 |
. . 3
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6 | oveq1 5902 |
. . 3
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7 | 5, 6 | eqeq12d 2204 |
. 2
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8 | ecovcom.4 |
. . . 4
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9 | ecovcom.5 |
. . . 4
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10 | opeq12 3795 |
. . . . 5
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11 | 10 | eceq1d 6594 |
. . . 4
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12 | 8, 9, 11 | mp2an 426 |
. . 3
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13 | ecovcom.2 |
. . 3
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14 | ecovcom.3 |
. . . 4
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15 | 14 | ancoms 268 |
. . 3
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16 | 12, 13, 15 | 3eqtr4a 2248 |
. 2
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17 | 1, 4, 7, 16 | 2ecoptocl 6648 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-xp 4650 df-cnv 4652 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fv 5243 df-ov 5898 df-ec 6560 df-qs 6564 |
This theorem is referenced by: (None) |
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