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Mirrors > Home > ILE Home > Th. List > reuind | Unicode version |
Description: Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.) |
Ref | Expression |
---|---|
reuind.1 | |
reuind.2 |
Ref | Expression |
---|---|
reuind |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reuind.2 | . . . . . . . 8 | |
2 | 1 | eleq1d 2106 | . . . . . . 7 |
3 | reuind.1 | . . . . . . 7 | |
4 | 2, 3 | anbi12d 442 | . . . . . 6 |
5 | 4 | cbvexv 1795 | . . . . 5 |
6 | r19.41v 2466 | . . . . . . 7 | |
7 | 6 | exbii 1496 | . . . . . 6 |
8 | rexcom4 2577 | . . . . . 6 | |
9 | risset 2352 | . . . . . . . 8 | |
10 | 9 | anbi1i 431 | . . . . . . 7 |
11 | 10 | exbii 1496 | . . . . . 6 |
12 | 7, 8, 11 | 3bitr4ri 202 | . . . . 5 |
13 | 5, 12 | bitri 173 | . . . 4 |
14 | eqeq2 2049 | . . . . . . . . . 10 | |
15 | 14 | imim2i 12 | . . . . . . . . 9 |
16 | bi2 121 | . . . . . . . . . . 11 | |
17 | 16 | imim2i 12 | . . . . . . . . . 10 |
18 | an31 498 | . . . . . . . . . . . 12 | |
19 | 18 | imbi1i 227 | . . . . . . . . . . 11 |
20 | impexp 250 | . . . . . . . . . . 11 | |
21 | impexp 250 | . . . . . . . . . . 11 | |
22 | 19, 20, 21 | 3bitr3i 199 | . . . . . . . . . 10 |
23 | 17, 22 | sylib 127 | . . . . . . . . 9 |
24 | 15, 23 | syl 14 | . . . . . . . 8 |
25 | 24 | 2alimi 1345 | . . . . . . 7 |
26 | 19.23v 1763 | . . . . . . . . . 10 | |
27 | an12 495 | . . . . . . . . . . . . . 14 | |
28 | eleq1 2100 | . . . . . . . . . . . . . . . 16 | |
29 | 28 | adantr 261 | . . . . . . . . . . . . . . 15 |
30 | 29 | pm5.32ri 428 | . . . . . . . . . . . . . 14 |
31 | 27, 30 | bitr4i 176 | . . . . . . . . . . . . 13 |
32 | 31 | exbii 1496 | . . . . . . . . . . . 12 |
33 | 19.42v 1786 | . . . . . . . . . . . 12 | |
34 | 32, 33 | bitri 173 | . . . . . . . . . . 11 |
35 | 34 | imbi1i 227 | . . . . . . . . . 10 |
36 | 26, 35 | bitri 173 | . . . . . . . . 9 |
37 | 36 | albii 1359 | . . . . . . . 8 |
38 | 19.21v 1753 | . . . . . . . 8 | |
39 | 37, 38 | bitri 173 | . . . . . . 7 |
40 | 25, 39 | sylib 127 | . . . . . 6 |
41 | 40 | expd 245 | . . . . 5 |
42 | 41 | reximdvai 2419 | . . . 4 |
43 | 13, 42 | syl5bi 141 | . . 3 |
44 | 43 | imp 115 | . 2 |
45 | pm4.24 375 | . . . . . . . . 9 | |
46 | 45 | biimpi 113 | . . . . . . . 8 |
47 | prth 326 | . . . . . . . 8 | |
48 | eqtr3 2059 | . . . . . . . 8 | |
49 | 46, 47, 48 | syl56 30 | . . . . . . 7 |
50 | 49 | alanimi 1348 | . . . . . 6 |
51 | 19.23v 1763 | . . . . . . . 8 | |
52 | 51 | biimpi 113 | . . . . . . 7 |
53 | 52 | com12 27 | . . . . . 6 |
54 | 50, 53 | syl5 28 | . . . . 5 |
55 | 54 | a1d 22 | . . . 4 |
56 | 55 | ralrimivv 2400 | . . 3 |
57 | 56 | adantl 262 | . 2 |
58 | eqeq1 2046 | . . . . 5 | |
59 | 58 | imbi2d 219 | . . . 4 |
60 | 59 | albidv 1705 | . . 3 |
61 | 60 | reu4 2735 | . 2 |
62 | 44, 57, 61 | 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 wral 2306 wrex 2307 wreu 2308 |
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-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-reu 2313 df-rmo 2314 df-v 2559 |
This theorem is referenced by: (None) |
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