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Mirrors > Home > NFE 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 2419 | . . . . . . 7 |
3 | reuind.1 | . . . . . . 7 | |
4 | 2, 3 | anbi12d 691 | . . . . . 6 |
5 | 4 | cbvexv 2003 | . . . . 5 |
6 | r19.41v 2764 | . . . . . . 7 | |
7 | 6 | exbii 1582 | . . . . . 6 |
8 | rexcom4 2878 | . . . . . 6 | |
9 | risset 2661 | . . . . . . . 8 | |
10 | 9 | anbi1i 676 | . . . . . . 7 |
11 | 10 | exbii 1582 | . . . . . 6 |
12 | 7, 8, 11 | 3bitr4ri 269 | . . . . 5 |
13 | 5, 12 | bitri 240 | . . . 4 |
14 | eqeq2 2362 | . . . . . . . . . 10 | |
15 | 14 | imim2i 13 | . . . . . . . . 9 |
16 | bi2 189 | . . . . . . . . . . 11 | |
17 | 16 | imim2i 13 | . . . . . . . . . 10 |
18 | an31 775 | . . . . . . . . . . . 12 | |
19 | 18 | imbi1i 315 | . . . . . . . . . . 11 |
20 | impexp 433 | . . . . . . . . . . 11 | |
21 | impexp 433 | . . . . . . . . . . 11 | |
22 | 19, 20, 21 | 3bitr3i 266 | . . . . . . . . . 10 |
23 | 17, 22 | sylib 188 | . . . . . . . . 9 |
24 | 15, 23 | syl 15 | . . . . . . . 8 |
25 | 24 | 2alimi 1560 | . . . . . . 7 |
26 | 19.23v 1891 | . . . . . . . . . 10 | |
27 | an12 772 | . . . . . . . . . . . . . 14 | |
28 | eleq1 2413 | . . . . . . . . . . . . . . . 16 | |
29 | 28 | adantr 451 | . . . . . . . . . . . . . . 15 |
30 | 29 | pm5.32ri 619 | . . . . . . . . . . . . . 14 |
31 | 27, 30 | bitr4i 243 | . . . . . . . . . . . . 13 |
32 | 31 | exbii 1582 | . . . . . . . . . . . 12 |
33 | 19.42v 1905 | . . . . . . . . . . . 12 | |
34 | 32, 33 | bitri 240 | . . . . . . . . . . 11 |
35 | 34 | imbi1i 315 | . . . . . . . . . 10 |
36 | 26, 35 | bitri 240 | . . . . . . . . 9 |
37 | 36 | albii 1566 | . . . . . . . 8 |
38 | 19.21v 1890 | . . . . . . . 8 | |
39 | 37, 38 | bitri 240 | . . . . . . 7 |
40 | 25, 39 | sylib 188 | . . . . . 6 |
41 | 40 | exp3a 425 | . . . . 5 |
42 | 41 | reximdvai 2724 | . . . 4 |
43 | 13, 42 | syl5bi 208 | . . 3 |
44 | 43 | imp 418 | . 2 |
45 | pm4.24 624 | . . . . . . . . 9 | |
46 | 45 | biimpi 186 | . . . . . . . 8 |
47 | prth 554 | . . . . . . . 8 | |
48 | eqtr3 2372 | . . . . . . . 8 | |
49 | 46, 47, 48 | syl56 30 | . . . . . . 7 |
50 | 49 | alanimi 1562 | . . . . . 6 |
51 | 19.23v 1891 | . . . . . . . 8 | |
52 | 51 | biimpi 186 | . . . . . . 7 |
53 | 52 | com12 27 | . . . . . 6 |
54 | 50, 53 | syl5 28 | . . . . 5 |
55 | 54 | a1d 22 | . . . 4 |
56 | 55 | ralrimivv 2705 | . . 3 |
57 | 56 | adantl 452 | . 2 |
58 | eqeq1 2359 | . . . . 5 | |
59 | 58 | imbi2d 307 | . . . 4 |
60 | 59 | albidv 1625 | . . 3 |
61 | 60 | reu4 3030 | . 2 |
62 | 44, 57, 61 | 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 wral 2614 wrex 2615 wreu 2616 |
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-nfc 2478 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-v 2861 |
This theorem is referenced by: (None) |
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