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Mirrors > Home > NFE 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 2003 | . . . . 5 |
3 | euind.1 | . . . . . . . . 9 | |
4 | 3 | isseti 2865 | . . . . . . . 8 |
5 | 4 | biantrur 492 | . . . . . . 7 |
6 | 5 | exbii 1582 | . . . . . 6 |
7 | 19.41v 1901 | . . . . . . . 8 | |
8 | 7 | exbii 1582 | . . . . . . 7 |
9 | excom 1741 | . . . . . . 7 | |
10 | 8, 9 | bitr3i 242 | . . . . . 6 |
11 | 6, 10 | bitri 240 | . . . . 5 |
12 | 2, 11 | bitri 240 | . . . 4 |
13 | eqeq2 2362 | . . . . . . . . 9 | |
14 | 13 | imim2i 13 | . . . . . . . 8 |
15 | bi2 189 | . . . . . . . . . 10 | |
16 | 15 | imim2i 13 | . . . . . . . . 9 |
17 | an31 775 | . . . . . . . . . . 11 | |
18 | 17 | imbi1i 315 | . . . . . . . . . 10 |
19 | impexp 433 | . . . . . . . . . 10 | |
20 | impexp 433 | . . . . . . . . . 10 | |
21 | 18, 19, 20 | 3bitr3i 266 | . . . . . . . . 9 |
22 | 16, 21 | sylib 188 | . . . . . . . 8 |
23 | 14, 22 | syl 15 | . . . . . . 7 |
24 | 23 | 2alimi 1560 | . . . . . 6 |
25 | 19.23v 1891 | . . . . . . . 8 | |
26 | 25 | albii 1566 | . . . . . . 7 |
27 | 19.21v 1890 | . . . . . . 7 | |
28 | 26, 27 | bitri 240 | . . . . . 6 |
29 | 24, 28 | sylib 188 | . . . . 5 |
30 | 29 | eximdv 1622 | . . . 4 |
31 | 12, 30 | syl5bi 208 | . . 3 |
32 | 31 | imp 418 | . 2 |
33 | pm4.24 624 | . . . . . . . 8 | |
34 | 33 | biimpi 186 | . . . . . . 7 |
35 | prth 554 | . . . . . . 7 | |
36 | eqtr3 2372 | . . . . . . 7 | |
37 | 34, 35, 36 | syl56 30 | . . . . . 6 |
38 | 37 | alanimi 1562 | . . . . 5 |
39 | 19.23v 1891 | . . . . . . 7 | |
40 | 39 | biimpi 186 | . . . . . 6 |
41 | 40 | com12 27 | . . . . 5 |
42 | 38, 41 | syl5 28 | . . . 4 |
43 | 42 | alrimivv 1632 | . . 3 |
44 | 43 | adantl 452 | . 2 |
45 | eqeq1 2359 | . . . . 5 | |
46 | 45 | imbi2d 307 | . . . 4 |
47 | 46 | albidv 1625 | . . 3 |
48 | 47 | eu4 2243 | . 2 |
49 | 32, 44, 48 | sylanbrc 645 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wa 358 wal 1540 wex 1541 wceq 1642 wcel 1710 weu 2204 cvv 2859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-v 2861 |
This theorem is referenced by: (None) |
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