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Mirrors > Home > ILE Home > Th. List > equveli | Unicode version |
Description: A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1641.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.) |
Ref | Expression |
---|---|
equveli |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | albiim 1376 |
. 2
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2 | ax12or 1403 |
. . 3
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3 | equequ1 1598 |
. . . . . . . . 9
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4 | equequ1 1598 |
. . . . . . . . 9
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5 | 3, 4 | imbi12d 223 |
. . . . . . . 8
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6 | 5 | sps 1430 |
. . . . . . 7
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7 | 6 | dral2 1619 |
. . . . . 6
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8 | equid 1589 |
. . . . . . . . 9
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9 | 8 | a1bi 232 |
. . . . . . . 8
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10 | 9 | biimpri 124 |
. . . . . . 7
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11 | 10 | sps 1430 |
. . . . . 6
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12 | 7, 11 | syl6bi 152 |
. . . . 5
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13 | 12 | adantrd 264 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | equequ1 1598 |
. . . . . . . . . 10
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15 | equequ1 1598 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 14, 15 | imbi12d 223 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 16 | sps 1430 |
. . . . . . . 8
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18 | 17 | dral1 1618 |
. . . . . . 7
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19 | equid 1589 |
. . . . . . . . 9
![]() ![]() ![]() ![]() | |
20 | ax-4 1400 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 19, 20 | mpi 15 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | equcomi 1592 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | 21, 22 | syl 14 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 18, 23 | syl6bi 152 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24 | adantld 263 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | hba1 1433 |
. . . . . . . . . 10
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27 | hbequid 1406 |
. . . . . . . . . . 11
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28 | 27 | a1i 9 |
. . . . . . . . . 10
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29 | ax-4 1400 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
30 | 26, 28, 29 | hbimd 1465 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 30 | a5i 1435 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | equtr 1595 |
. . . . . . . . . 10
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33 | ax-8 1395 |
. . . . . . . . . 10
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34 | 32, 33 | imim12d 68 |
. . . . . . . . 9
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35 | 34 | ax-gen 1338 |
. . . . . . . 8
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36 | 19.26 1370 |
. . . . . . . . 9
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37 | spimth 1623 |
. . . . . . . . 9
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38 | 36, 37 | sylbir 125 |
. . . . . . . 8
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39 | 31, 35, 38 | sylancl 392 |
. . . . . . 7
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40 | 8, 39 | mpii 39 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
41 | 40 | adantrd 264 |
. . . . 5
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42 | 25, 41 | jaoi 636 |
. . . 4
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43 | 13, 42 | jaoi 636 |
. . 3
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44 | 2, 43 | ax-mp 7 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
45 | 1, 44 | sylbi 114 |
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 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-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: (None) |
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