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Mirrors > Home > MPE Home > Th. List > euind | Structured version Visualization version 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 2275 | . . . . 5 |
3 | euind.1 | . . . . . . . . 9 | |
4 | 3 | isseti 3209 | . . . . . . . 8 |
5 | 4 | biantrur 527 | . . . . . . 7 |
6 | 5 | exbii 1774 | . . . . . 6 |
7 | 19.41v 1914 | . . . . . . 7 | |
8 | 7 | exbii 1774 | . . . . . 6 |
9 | excom 2042 | . . . . . 6 | |
10 | 6, 8, 9 | 3bitr2i 288 | . . . . 5 |
11 | 2, 10 | bitri 264 | . . . 4 |
12 | eqeq2 2633 | . . . . . . . . 9 | |
13 | 12 | imim2i 16 | . . . . . . . 8 |
14 | biimpr 210 | . . . . . . . . . 10 | |
15 | 14 | imim2i 16 | . . . . . . . . 9 |
16 | an31 841 | . . . . . . . . . . 11 | |
17 | 16 | imbi1i 339 | . . . . . . . . . 10 |
18 | impexp 462 | . . . . . . . . . 10 | |
19 | impexp 462 | . . . . . . . . . 10 | |
20 | 17, 18, 19 | 3bitr3i 290 | . . . . . . . . 9 |
21 | 15, 20 | sylib 208 | . . . . . . . 8 |
22 | 13, 21 | syl 17 | . . . . . . 7 |
23 | 22 | 2alimi 1740 | . . . . . 6 |
24 | 19.23v 1902 | . . . . . . . 8 | |
25 | 24 | albii 1747 | . . . . . . 7 |
26 | 19.21v 1868 | . . . . . . 7 | |
27 | 25, 26 | bitri 264 | . . . . . 6 |
28 | 23, 27 | sylib 208 | . . . . 5 |
29 | 28 | eximdv 1846 | . . . 4 |
30 | 11, 29 | syl5bi 232 | . . 3 |
31 | 30 | imp 445 | . 2 |
32 | pm4.24 675 | . . . . . . . . 9 | |
33 | 32 | biimpi 206 | . . . . . . . 8 |
34 | prth 595 | . . . . . . . 8 | |
35 | eqtr3 2643 | . . . . . . . 8 | |
36 | 33, 34, 35 | syl56 36 | . . . . . . 7 |
37 | 36 | alanimi 1744 | . . . . . 6 |
38 | 19.23v 1902 | . . . . . 6 | |
39 | 37, 38 | sylib 208 | . . . . 5 |
40 | 39 | com12 32 | . . . 4 |
41 | 40 | alrimivv 1856 | . . 3 |
42 | 41 | adantl 482 | . 2 |
43 | eqeq1 2626 | . . . . 5 | |
44 | 43 | imbi2d 330 | . . . 4 |
45 | 44 | albidv 1849 | . . 3 |
46 | 45 | eu4 2518 | . 2 |
47 | 31, 42, 46 | sylanbrc 698 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wal 1481 wceq 1483 wex 1704 wcel 1990 weu 2470 cvv 3200 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-v 3202 |
This theorem is referenced by: (None) |
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