Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | albiim 1376 | . 2 | |
2 | ax12or 1403 | . . 3 | |
3 | equequ1 1598 | . . . . . . . . 9 | |
4 | equequ1 1598 | . . . . . . . . 9 | |
5 | 3, 4 | imbi12d 223 | . . . . . . . 8 |
6 | 5 | sps 1430 | . . . . . . 7 |
7 | 6 | dral2 1619 | . . . . . 6 |
8 | equid 1589 | . . . . . . . . 9 | |
9 | 8 | a1bi 232 | . . . . . . . 8 |
10 | 9 | biimpri 124 | . . . . . . 7 |
11 | 10 | sps 1430 | . . . . . 6 |
12 | 7, 11 | syl6bi 152 | . . . . 5 |
13 | 12 | adantrd 264 | . . . 4 |
14 | equequ1 1598 | . . . . . . . . . 10 | |
15 | equequ1 1598 | . . . . . . . . . 10 | |
16 | 14, 15 | imbi12d 223 | . . . . . . . . 9 |
17 | 16 | sps 1430 | . . . . . . . 8 |
18 | 17 | dral1 1618 | . . . . . . 7 |
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 | |
27 | hbequid 1406 | . . . . . . . . . . 11 | |
28 | 27 | a1i 9 | . . . . . . . . . 10 |
29 | ax-4 1400 | . . . . . . . . . 10 | |
30 | 26, 28, 29 | hbimd 1465 | . . . . . . . . 9 |
31 | 30 | a5i 1435 | . . . . . . . 8 |
32 | equtr 1595 | . . . . . . . . . 10 | |
33 | ax-8 1395 | . . . . . . . . . 10 | |
34 | 32, 33 | imim12d 68 | . . . . . . . . 9 |
35 | 34 | ax-gen 1338 | . . . . . . . 8 |
36 | 19.26 1370 | . . . . . . . . 9 | |
37 | spimth 1623 | . . . . . . . . 9 | |
38 | 36, 37 | sylbir 125 | . . . . . . . 8 |
39 | 31, 35, 38 | sylancl 392 | . . . . . . 7 |
40 | 8, 39 | mpii 39 | . . . . . 6 |
41 | 40 | adantrd 264 | . . . . 5 |
42 | 25, 41 | jaoi 636 | . . . 4 |
43 | 13, 42 | jaoi 636 | . . 3 |
44 | 2, 43 | ax-mp 7 | . 2 |
45 | 1, 44 | sylbi 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wo 629 wal 1241 wceq 1243 |
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) |
Copyright terms: Public domain | W3C validator |