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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 1756.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.) |
Ref | Expression |
---|---|
equveli |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | albiim 1485 | . 2 | |
2 | ax12or 1506 | . . 3 | |
3 | equequ1 1710 | . . . . . . . . 9 | |
4 | equequ1 1710 | . . . . . . . . 9 | |
5 | 3, 4 | imbi12d 234 | . . . . . . . 8 |
6 | 5 | sps 1535 | . . . . . . 7 |
7 | 6 | dral2 1729 | . . . . . 6 |
8 | equid 1699 | . . . . . . . . 9 | |
9 | 8 | a1bi 243 | . . . . . . . 8 |
10 | 9 | biimpri 133 | . . . . . . 7 |
11 | 10 | sps 1535 | . . . . . 6 |
12 | 7, 11 | syl6bi 163 | . . . . 5 |
13 | 12 | adantrd 279 | . . . 4 |
14 | equequ1 1710 | . . . . . . . . . 10 | |
15 | equequ1 1710 | . . . . . . . . . 10 | |
16 | 14, 15 | imbi12d 234 | . . . . . . . . 9 |
17 | 16 | sps 1535 | . . . . . . . 8 |
18 | 17 | dral1 1728 | . . . . . . 7 |
19 | equid 1699 | . . . . . . . . 9 | |
20 | ax-4 1508 | . . . . . . . . 9 | |
21 | 19, 20 | mpi 15 | . . . . . . . 8 |
22 | equcomi 1702 | . . . . . . . 8 | |
23 | 21, 22 | syl 14 | . . . . . . 7 |
24 | 18, 23 | syl6bi 163 | . . . . . 6 |
25 | 24 | adantld 278 | . . . . 5 |
26 | hba1 1538 | . . . . . . . . . 10 | |
27 | hbequid 1511 | . . . . . . . . . . 11 | |
28 | 27 | a1i 9 | . . . . . . . . . 10 |
29 | ax-4 1508 | . . . . . . . . . 10 | |
30 | 26, 28, 29 | hbimd 1571 | . . . . . . . . 9 |
31 | 30 | a5i 1541 | . . . . . . . 8 |
32 | equtr 1707 | . . . . . . . . . 10 | |
33 | ax-8 1502 | . . . . . . . . . 10 | |
34 | 32, 33 | imim12d 74 | . . . . . . . . 9 |
35 | 34 | ax-gen 1447 | . . . . . . . 8 |
36 | 19.26 1479 | . . . . . . . . 9 | |
37 | spimth 1733 | . . . . . . . . 9 | |
38 | 36, 37 | sylbir 135 | . . . . . . . 8 |
39 | 31, 35, 38 | sylancl 413 | . . . . . . 7 |
40 | 8, 39 | mpii 44 | . . . . . 6 |
41 | 40 | adantrd 279 | . . . . 5 |
42 | 25, 41 | jaoi 716 | . . . 4 |
43 | 13, 42 | jaoi 716 | . . 3 |
44 | 2, 43 | ax-mp 5 | . 2 |
45 | 1, 44 | sylbi 121 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wb 105 wo 708 wal 1351 wceq 1353 |
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 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: (None) |
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