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Mirrors > Home > ILE Home > Th. List > euind | Unicode version |
Description: Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.) |
Ref | Expression |
---|---|
euind.1 | |
euind.2 | |
euind.3 |
Ref | Expression |
---|---|
euind |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | euind.2 | . . . . . 6 | |
2 | 1 | cbvexv 1795 | . . . . 5 |
3 | euind.1 | . . . . . . . . 9 | |
4 | 3 | isseti 2563 | . . . . . . . 8 |
5 | 4 | biantrur 287 | . . . . . . 7 |
6 | 5 | exbii 1496 | . . . . . 6 |
7 | 19.41v 1782 | . . . . . . 7 | |
8 | 7 | exbii 1496 | . . . . . 6 |
9 | excom 1554 | . . . . . 6 | |
10 | 6, 8, 9 | 3bitr2i 197 | . . . . 5 |
11 | 2, 10 | bitri 173 | . . . 4 |
12 | eqeq2 2049 | . . . . . . . . 9 | |
13 | 12 | imim2i 12 | . . . . . . . 8 |
14 | bi2 121 | . . . . . . . . . 10 | |
15 | 14 | imim2i 12 | . . . . . . . . 9 |
16 | an31 498 | . . . . . . . . . . 11 | |
17 | 16 | imbi1i 227 | . . . . . . . . . 10 |
18 | impexp 250 | . . . . . . . . . 10 | |
19 | impexp 250 | . . . . . . . . . 10 | |
20 | 17, 18, 19 | 3bitr3i 199 | . . . . . . . . 9 |
21 | 15, 20 | sylib 127 | . . . . . . . 8 |
22 | 13, 21 | syl 14 | . . . . . . 7 |
23 | 22 | 2alimi 1345 | . . . . . 6 |
24 | 19.23v 1763 | . . . . . . . 8 | |
25 | 24 | albii 1359 | . . . . . . 7 |
26 | 19.21v 1753 | . . . . . . 7 | |
27 | 25, 26 | bitri 173 | . . . . . 6 |
28 | 23, 27 | sylib 127 | . . . . 5 |
29 | 28 | eximdv 1760 | . . . 4 |
30 | 11, 29 | syl5bi 141 | . . 3 |
31 | 30 | imp 115 | . 2 |
32 | pm4.24 375 | . . . . . . . 8 | |
33 | 32 | biimpi 113 | . . . . . . 7 |
34 | prth 326 | . . . . . . 7 | |
35 | eqtr3 2059 | . . . . . . 7 | |
36 | 33, 34, 35 | syl56 30 | . . . . . 6 |
37 | 36 | alanimi 1348 | . . . . 5 |
38 | 19.23v 1763 | . . . . . . 7 | |
39 | 38 | biimpi 113 | . . . . . 6 |
40 | 39 | com12 27 | . . . . 5 |
41 | 37, 40 | syl5 28 | . . . 4 |
42 | 41 | alrimivv 1755 | . . 3 |
43 | 42 | adantl 262 | . 2 |
44 | eqeq1 2046 | . . . . 5 | |
45 | 44 | imbi2d 219 | . . . 4 |
46 | 45 | albidv 1705 | . . 3 |
47 | 46 | eu4 1962 | . 2 |
48 | 31, 43, 47 | sylanbrc 394 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wal 1241 wceq 1243 wex 1381 wcel 1393 weu 1900 cvv 2557 |
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-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-v 2559 |
This theorem is referenced by: (None) |
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