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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 1731.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.) |
Ref | Expression |
---|---|
equveli |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | albiim 1463 | . 2 | |
2 | ax12or 1490 | . . 3 | |
3 | equequ1 1688 | . . . . . . . . 9 | |
4 | equequ1 1688 | . . . . . . . . 9 | |
5 | 3, 4 | imbi12d 233 | . . . . . . . 8 |
6 | 5 | sps 1517 | . . . . . . 7 |
7 | 6 | dral2 1709 | . . . . . 6 |
8 | equid 1677 | . . . . . . . . 9 | |
9 | 8 | a1bi 242 | . . . . . . . 8 |
10 | 9 | biimpri 132 | . . . . . . 7 |
11 | 10 | sps 1517 | . . . . . 6 |
12 | 7, 11 | syl6bi 162 | . . . . 5 |
13 | 12 | adantrd 277 | . . . 4 |
14 | equequ1 1688 | . . . . . . . . . 10 | |
15 | equequ1 1688 | . . . . . . . . . 10 | |
16 | 14, 15 | imbi12d 233 | . . . . . . . . 9 |
17 | 16 | sps 1517 | . . . . . . . 8 |
18 | 17 | dral1 1708 | . . . . . . 7 |
19 | equid 1677 | . . . . . . . . 9 | |
20 | ax-4 1487 | . . . . . . . . 9 | |
21 | 19, 20 | mpi 15 | . . . . . . . 8 |
22 | equcomi 1680 | . . . . . . . 8 | |
23 | 21, 22 | syl 14 | . . . . . . 7 |
24 | 18, 23 | syl6bi 162 | . . . . . 6 |
25 | 24 | adantld 276 | . . . . 5 |
26 | hba1 1520 | . . . . . . . . . 10 | |
27 | hbequid 1493 | . . . . . . . . . . 11 | |
28 | 27 | a1i 9 | . . . . . . . . . 10 |
29 | ax-4 1487 | . . . . . . . . . 10 | |
30 | 26, 28, 29 | hbimd 1552 | . . . . . . . . 9 |
31 | 30 | a5i 1522 | . . . . . . . 8 |
32 | equtr 1685 | . . . . . . . . . 10 | |
33 | ax-8 1482 | . . . . . . . . . 10 | |
34 | 32, 33 | imim12d 74 | . . . . . . . . 9 |
35 | 34 | ax-gen 1425 | . . . . . . . 8 |
36 | 19.26 1457 | . . . . . . . . 9 | |
37 | spimth 1713 | . . . . . . . . 9 | |
38 | 36, 37 | sylbir 134 | . . . . . . . 8 |
39 | 31, 35, 38 | sylancl 409 | . . . . . . 7 |
40 | 8, 39 | mpii 44 | . . . . . 6 |
41 | 40 | adantrd 277 | . . . . 5 |
42 | 25, 41 | jaoi 705 | . . . 4 |
43 | 13, 42 | jaoi 705 | . . 3 |
44 | 2, 43 | ax-mp 5 | . 2 |
45 | 1, 44 | sylbi 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 697 wal 1329 wceq 1331 |
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-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: (None) |
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