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Theorem 2eu4 2371
 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 2eu1 2368 for a condition under which the naive definition holds and 2exeu 2365 for a one-way implication. See 2eu5 2372 and 2eu8 2375 for alternate definitions. (Contributed by NM, 3-Dec-2001.)
Assertion
Ref Expression
2eu4
Distinct variable groups:   ,,,   ,,
Allowed substitution hints:   (,)

Proof of Theorem 2eu4
StepHypRef Expression
1 nfv 1631 . . . 4
21eu3 2314 . . 3
3 nfv 1631 . . . 4
43eu3 2314 . . 3
52, 4anbi12i 680 . 2
6 an4 799 . 2
7 excom 1759 . . . . 5
87anbi2i 677 . . . 4
9 anidm 627 . . . 4
108, 9bitri 242 . . 3
11 19.26 1605 . . . . . . . 8
12 nfa1 1809 . . . . . . . . . . 11
131219.3 1794 . . . . . . . . . 10
1413anbi2i 677 . . . . . . . . 9
15 jcab 835 . . . . . . . . . . . . 13
1615albii 1576 . . . . . . . . . . . 12
17 19.26 1605 . . . . . . . . . . . 12
1816, 17bitri 242 . . . . . . . . . . 11
1918albii 1576 . . . . . . . . . 10
20 19.26 1605 . . . . . . . . . 10
2119, 20bitri 242 . . . . . . . . 9
2214, 21bitr4i 245 . . . . . . . 8
2311, 22bitr2i 243 . . . . . . 7
24 19.26 1605 . . . . . . . . 9
25 nfa1 1809 . . . . . . . . . . 11
262519.3 1794 . . . . . . . . . 10
27 alcom 1755 . . . . . . . . . 10
2826, 27anbi12i 680 . . . . . . . . 9
2924, 28bitri 242 . . . . . . . 8
3029albii 1576 . . . . . . 7
3123, 30bitr4i 245 . . . . . 6
32 19.23v 1918 . . . . . . . 8
33 19.23v 1918 . . . . . . . 8
3432, 33anbi12i 680 . . . . . . 7
35342albii 1577 . . . . . 6
36 nfe1 1750 . . . . . . . 8
37 nfv 1631 . . . . . . . 8
3836, 37nfim 1835 . . . . . . 7
39 nfe1 1750 . . . . . . . 8
40 nfv 1631 . . . . . . . 8
4139, 40nfim 1835 . . . . . . 7
4238, 41aaan 1910 . . . . . 6
4331, 35, 423bitri 264 . . . . 5
44432exbii 1594 . . . 4
45 eeanv 1941 . . . 4
4644, 45bitr2i 243 . . 3
4710, 46anbi12i 680 . 2
485, 6, 473bitri 264 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360  wal 1550  wex 1551  weu 2288 This theorem is referenced by:  2eu5  2372  2eu6  2373 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292
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