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Mirrors > Home > ILE Home > Th. List > 2eu4 | Unicode version |
Description: This theorem provides us with a definition of double existential uniqueness ("exactly one and exactly one "). Naively one might think (incorrectly) that it could be defined by . See 2exeu 2033 for a one-way implication. (Contributed by NM, 3-Dec-2001.) |
Ref | Expression |
---|---|
2eu4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-17 1459 | . . . 4 | |
2 | 1 | eu3h 1986 | . . 3 |
3 | ax-17 1459 | . . . 4 | |
4 | 3 | eu3h 1986 | . . 3 |
5 | 2, 4 | anbi12i 447 | . 2 |
6 | an4 550 | . 2 | |
7 | excom 1594 | . . . . 5 | |
8 | 7 | anbi2i 444 | . . . 4 |
9 | anidm 388 | . . . 4 | |
10 | 8, 9 | bitri 182 | . . 3 |
11 | hba1 1473 | . . . . . . . . . 10 | |
12 | 11 | 19.3h 1485 | . . . . . . . . 9 |
13 | 12 | anbi2i 444 | . . . . . . . 8 |
14 | 19.26 1410 | . . . . . . . 8 | |
15 | jcab 567 | . . . . . . . . . . . 12 | |
16 | 15 | albii 1399 | . . . . . . . . . . 11 |
17 | 19.26 1410 | . . . . . . . . . . 11 | |
18 | 16, 17 | bitri 182 | . . . . . . . . . 10 |
19 | 18 | albii 1399 | . . . . . . . . 9 |
20 | 19.26 1410 | . . . . . . . . 9 | |
21 | 19, 20 | bitri 182 | . . . . . . . 8 |
22 | 13, 14, 21 | 3bitr4ri 211 | . . . . . . 7 |
23 | 19.26 1410 | . . . . . . . . 9 | |
24 | hba1 1473 | . . . . . . . . . . 11 | |
25 | 24 | 19.3h 1485 | . . . . . . . . . 10 |
26 | alcom 1407 | . . . . . . . . . 10 | |
27 | 25, 26 | anbi12i 447 | . . . . . . . . 9 |
28 | 23, 27 | bitri 182 | . . . . . . . 8 |
29 | 28 | albii 1399 | . . . . . . 7 |
30 | 22, 29 | bitr4i 185 | . . . . . 6 |
31 | 19.23v 1804 | . . . . . . . 8 | |
32 | 19.23v 1804 | . . . . . . . 8 | |
33 | 31, 32 | anbi12i 447 | . . . . . . 7 |
34 | 33 | 2albii 1400 | . . . . . 6 |
35 | hbe1 1424 | . . . . . . . 8 | |
36 | ax-17 1459 | . . . . . . . 8 | |
37 | 35, 36 | hbim 1477 | . . . . . . 7 |
38 | hbe1 1424 | . . . . . . . 8 | |
39 | ax-17 1459 | . . . . . . . 8 | |
40 | 38, 39 | hbim 1477 | . . . . . . 7 |
41 | 37, 40 | aaanh 1518 | . . . . . 6 |
42 | 30, 34, 41 | 3bitri 204 | . . . . 5 |
43 | 42 | 2exbii 1537 | . . . 4 |
44 | eeanv 1848 | . . . 4 | |
45 | 43, 44 | bitr2i 183 | . . 3 |
46 | 10, 45 | anbi12i 447 | . 2 |
47 | 5, 6, 46 | 3bitri 204 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wal 1282 wex 1421 weu 1941 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-eu 1944 |
This theorem is referenced by: (None) |
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