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Theorem 2eu4 1990
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 1989 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 1416 . . . 4
21eu3h 1942 . . 3
3 ax-17 1416 . . . 4
43eu3h 1942 . . 3
52, 4anbi12i 433 . 2
6 an4 520 . 2
7 excom 1551 . . . . 5
87anbi2i 430 . . . 4
9 anidm 376 . . . 4
108, 9bitri 173 . . 3
11 hba1 1430 . . . . . . . . . 10
121119.3h 1442 . . . . . . . . 9
1312anbi2i 430 . . . . . . . 8
14 19.26 1367 . . . . . . . 8
15 jcab 535 . . . . . . . . . . . 12
1615albii 1356 . . . . . . . . . . 11
17 19.26 1367 . . . . . . . . . . 11
1816, 17bitri 173 . . . . . . . . . 10
1918albii 1356 . . . . . . . . 9
20 19.26 1367 . . . . . . . . 9
2119, 20bitri 173 . . . . . . . 8
2213, 14, 213bitr4ri 202 . . . . . . 7
23 19.26 1367 . . . . . . . . 9
24 hba1 1430 . . . . . . . . . . 11
252419.3h 1442 . . . . . . . . . 10
26 alcom 1364 . . . . . . . . . 10
2725, 26anbi12i 433 . . . . . . . . 9
2823, 27bitri 173 . . . . . . . 8
2928albii 1356 . . . . . . 7
3022, 29bitr4i 176 . . . . . 6
31 19.23v 1760 . . . . . . . 8
32 19.23v 1760 . . . . . . . 8
3331, 32anbi12i 433 . . . . . . 7
34332albii 1357 . . . . . 6
35 hbe1 1381 . . . . . . . 8
36 ax-17 1416 . . . . . . . 8
3735, 36hbim 1434 . . . . . . 7
38 hbe1 1381 . . . . . . . 8
39 ax-17 1416 . . . . . . . 8
4038, 39hbim 1434 . . . . . . 7
4137, 40aaanh 1475 . . . . . 6
4230, 34, 413bitri 195 . . . . 5
43422exbii 1494 . . . 4
44 eeanv 1804 . . . 4
4543, 44bitr2i 174 . . 3
4610, 45anbi12i 433 . 2
475, 6, 463bitri 195 1
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98  wal 1240  wex 1378  weu 1897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900
This theorem is referenced by: (None)
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