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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 2186 | . . . . . . 7 |
3 | reuind.1 | . . . . . . 7 | |
4 | 2, 3 | anbi12d 464 | . . . . . 6 |
5 | 4 | cbvexv 1872 | . . . . 5 |
6 | r19.41v 2564 | . . . . . . 7 | |
7 | 6 | exbii 1569 | . . . . . 6 |
8 | rexcom4 2683 | . . . . . 6 | |
9 | risset 2440 | . . . . . . . 8 | |
10 | 9 | anbi1i 453 | . . . . . . 7 |
11 | 10 | exbii 1569 | . . . . . 6 |
12 | 7, 8, 11 | 3bitr4ri 212 | . . . . 5 |
13 | 5, 12 | bitri 183 | . . . 4 |
14 | eqeq2 2127 | . . . . . . . . . 10 | |
15 | 14 | imim2i 12 | . . . . . . . . 9 |
16 | bi2 129 | . . . . . . . . . . 11 | |
17 | 16 | imim2i 12 | . . . . . . . . . 10 |
18 | an31 538 | . . . . . . . . . . . 12 | |
19 | 18 | imbi1i 237 | . . . . . . . . . . 11 |
20 | impexp 261 | . . . . . . . . . . 11 | |
21 | impexp 261 | . . . . . . . . . . 11 | |
22 | 19, 20, 21 | 3bitr3i 209 | . . . . . . . . . 10 |
23 | 17, 22 | sylib 121 | . . . . . . . . 9 |
24 | 15, 23 | syl 14 | . . . . . . . 8 |
25 | 24 | 2alimi 1417 | . . . . . . 7 |
26 | 19.23v 1839 | . . . . . . . . . 10 | |
27 | an12 535 | . . . . . . . . . . . . . 14 | |
28 | eleq1 2180 | . . . . . . . . . . . . . . . 16 | |
29 | 28 | adantr 274 | . . . . . . . . . . . . . . 15 |
30 | 29 | pm5.32ri 450 | . . . . . . . . . . . . . 14 |
31 | 27, 30 | bitr4i 186 | . . . . . . . . . . . . 13 |
32 | 31 | exbii 1569 | . . . . . . . . . . . 12 |
33 | 19.42v 1862 | . . . . . . . . . . . 12 | |
34 | 32, 33 | bitri 183 | . . . . . . . . . . 11 |
35 | 34 | imbi1i 237 | . . . . . . . . . 10 |
36 | 26, 35 | bitri 183 | . . . . . . . . 9 |
37 | 36 | albii 1431 | . . . . . . . 8 |
38 | 19.21v 1829 | . . . . . . . 8 | |
39 | 37, 38 | bitri 183 | . . . . . . 7 |
40 | 25, 39 | sylib 121 | . . . . . 6 |
41 | 40 | expd 256 | . . . . 5 |
42 | 41 | reximdvai 2509 | . . . 4 |
43 | 13, 42 | syl5bi 151 | . . 3 |
44 | 43 | imp 123 | . 2 |
45 | pm4.24 392 | . . . . . . . . 9 | |
46 | 45 | biimpi 119 | . . . . . . . 8 |
47 | anim12 341 | . . . . . . . 8 | |
48 | eqtr3 2137 | . . . . . . . 8 | |
49 | 46, 47, 48 | syl56 34 | . . . . . . 7 |
50 | 49 | alanimi 1420 | . . . . . 6 |
51 | 19.23v 1839 | . . . . . . . 8 | |
52 | 51 | biimpi 119 | . . . . . . 7 |
53 | 52 | com12 30 | . . . . . 6 |
54 | 50, 53 | syl5 32 | . . . . 5 |
55 | 54 | a1d 22 | . . . 4 |
56 | 55 | ralrimivv 2490 | . . 3 |
57 | 56 | adantl 275 | . 2 |
58 | eqeq1 2124 | . . . . 5 | |
59 | 58 | imbi2d 229 | . . . 4 |
60 | 59 | albidv 1780 | . . 3 |
61 | 60 | reu4 2851 | . 2 |
62 | 44, 57, 61 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1314 wceq 1316 wex 1453 wcel 1465 wral 2393 wrex 2394 wreu 2395 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-v 2662 |
This theorem is referenced by: (None) |
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