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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 1758.) (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 1487 |
. 2
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2 | ax12or 1508 |
. . 3
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3 | equequ1 1712 |
. . . . . . . . 9
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4 | equequ1 1712 |
. . . . . . . . 9
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5 | 3, 4 | imbi12d 234 |
. . . . . . . 8
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6 | 5 | sps 1537 |
. . . . . . 7
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7 | 6 | dral2 1731 |
. . . . . 6
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8 | equid 1701 |
. . . . . . . . 9
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9 | 8 | a1bi 243 |
. . . . . . . 8
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10 | 9 | biimpri 133 |
. . . . . . 7
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11 | 10 | sps 1537 |
. . . . . 6
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12 | 7, 11 | syl6bi 163 |
. . . . 5
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13 | 12 | adantrd 279 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | equequ1 1712 |
. . . . . . . . . 10
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15 | equequ1 1712 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 14, 15 | imbi12d 234 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 16 | sps 1537 |
. . . . . . . 8
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18 | 17 | dral1 1730 |
. . . . . . 7
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19 | equid 1701 |
. . . . . . . . 9
![]() ![]() ![]() ![]() | |
20 | ax-4 1510 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 19, 20 | mpi 15 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | equcomi 1704 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | 21, 22 | syl 14 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 18, 23 | syl6bi 163 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24 | adantld 278 |
. . . . 5
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26 | hba1 1540 |
. . . . . . . . . 10
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27 | hbequid 1513 |
. . . . . . . . . . 11
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28 | 27 | a1i 9 |
. . . . . . . . . 10
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29 | ax-4 1510 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
30 | 26, 28, 29 | hbimd 1573 |
. . . . . . . . 9
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31 | 30 | a5i 1543 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | equtr 1709 |
. . . . . . . . . 10
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33 | ax-8 1504 |
. . . . . . . . . 10
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34 | 32, 33 | imim12d 74 |
. . . . . . . . 9
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35 | 34 | ax-gen 1449 |
. . . . . . . 8
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36 | 19.26 1481 |
. . . . . . . . 9
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37 | spimth 1735 |
. . . . . . . . 9
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38 | 36, 37 | sylbir 135 |
. . . . . . . 8
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39 | 31, 35, 38 | sylancl 413 |
. . . . . . 7
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40 | 8, 39 | mpii 44 |
. . . . . 6
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41 | 40 | adantrd 279 |
. . . . 5
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42 | 25, 41 | jaoi 716 |
. . . 4
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43 | 13, 42 | jaoi 716 |
. . 3
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44 | 2, 43 | ax-mp 5 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
45 | 1, 44 | sylbi 121 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: (None) |
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