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Theorem 2eu4 2099
 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 2098 for a one-way implication. (Contributed by NM, 3-Dec-2001.)
Assertion
Ref Expression
2eu4
Distinct variable groups:   ,,,   ,,
Allowed substitution hints:   (,)

Proof of Theorem 2eu4
StepHypRef Expression
1 ax-17 1506 . . . 4
21eu3h 2051 . . 3
3 ax-17 1506 . . . 4
43eu3h 2051 . . 3
52, 4anbi12i 456 . 2
6 an4 576 . 2
7 excom 1644 . . . . 5
87anbi2i 453 . . . 4
9 anidm 394 . . . 4
108, 9bitri 183 . . 3
11 hba1 1520 . . . . . . . . . 10
121119.3h 1533 . . . . . . . . 9
1312anbi2i 453 . . . . . . . 8
14 19.26 1461 . . . . . . . 8
15 jcab 593 . . . . . . . . . . . 12
1615albii 1450 . . . . . . . . . . 11
17 19.26 1461 . . . . . . . . . . 11
1816, 17bitri 183 . . . . . . . . . 10
1918albii 1450 . . . . . . . . 9
20 19.26 1461 . . . . . . . . 9
2119, 20bitri 183 . . . . . . . 8
2213, 14, 213bitr4ri 212 . . . . . . 7
23 19.26 1461 . . . . . . . . 9
24 hba1 1520 . . . . . . . . . . 11
252419.3h 1533 . . . . . . . . . 10
26 alcom 1458 . . . . . . . . . 10
2725, 26anbi12i 456 . . . . . . . . 9
2823, 27bitri 183 . . . . . . . 8
2928albii 1450 . . . . . . 7
3022, 29bitr4i 186 . . . . . 6
31 19.23v 1863 . . . . . . . 8
32 19.23v 1863 . . . . . . . 8
3331, 32anbi12i 456 . . . . . . 7
34332albii 1451 . . . . . 6
35 hbe1 1475 . . . . . . . 8
36 ax-17 1506 . . . . . . . 8
3735, 36hbim 1525 . . . . . . 7
38 hbe1 1475 . . . . . . . 8
39 ax-17 1506 . . . . . . . 8
4038, 39hbim 1525 . . . . . . 7
4137, 40aaanh 1566 . . . . . 6
4230, 34, 413bitri 205 . . . . 5
43422exbii 1586 . . . 4
44 eeanv 1912 . . . 4
4543, 44bitr2i 184 . . 3
4610, 45anbi12i 456 . 2
475, 6, 463bitri 205 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104  wal 1333  wex 1472  weu 2006 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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515 This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-eu 2009 This theorem is referenced by: (None)
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