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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 6390 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 5651 |
. . 3
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3 | oveq2 5652 |
. . 3
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4 | 2, 3 | eqeq12d 2102 |
. 2
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5 | oveq2 5652 |
. . 3
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6 | oveq1 5651 |
. . 3
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7 | 5, 6 | eqeq12d 2102 |
. 2
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8 | ecovcom.4 |
. . . 4
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9 | ecovcom.5 |
. . . 4
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10 | opeq12 3622 |
. . . . 5
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11 | 10 | eceq1d 6318 |
. . . 4
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12 | 8, 9, 11 | mp2an 417 |
. . 3
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13 | ecovcom.2 |
. . 3
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14 | ecovcom.3 |
. . . 4
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15 | 14 | ancoms 264 |
. . 3
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16 | 12, 13, 15 | 3eqtr4a 2146 |
. 2
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17 | 1, 4, 7, 16 | 2ecoptocl 6370 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3955 ax-pow 4007 ax-pr 4034 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-pw 3429 df-sn 3450 df-pr 3451 df-op 3453 df-uni 3652 df-br 3844 df-opab 3898 df-xp 4442 df-cnv 4444 df-dm 4446 df-rn 4447 df-res 4448 df-ima 4449 df-iota 4975 df-fv 5018 df-ov 5647 df-ec 6284 df-qs 6288 |
This theorem is referenced by: (None) |
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