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Theorem 2eu4 2287
 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 2284 for a condition under which the naive definition holds and 2exeu 2281 for a one-way implication. See 2eu5 2288 and 2eu8 2291 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 1619 . . . 4
21eu3 2230 . . 3
3 nfv 1619 . . . 4
43eu3 2230 . . 3
52, 4anbi12i 678 . 2
6 an4 797 . 2
7 excom 1741 . . . . 5
87anbi2i 675 . . . 4
9 anidm 625 . . . 4
108, 9bitri 240 . . 3
11 19.26 1593 . . . . . . . 8
12 nfa1 1788 . . . . . . . . . . 11
131219.3 1785 . . . . . . . . . 10
1413anbi2i 675 . . . . . . . . 9
15 jcab 833 . . . . . . . . . . . . 13
1615albii 1566 . . . . . . . . . . . 12
17 19.26 1593 . . . . . . . . . . . 12
1816, 17bitri 240 . . . . . . . . . . 11
1918albii 1566 . . . . . . . . . 10
20 19.26 1593 . . . . . . . . . 10
2119, 20bitri 240 . . . . . . . . 9
2214, 21bitr4i 243 . . . . . . . 8
2311, 22bitr2i 241 . . . . . . 7
24 19.26 1593 . . . . . . . . 9
25 nfa1 1788 . . . . . . . . . . 11
262519.3 1785 . . . . . . . . . 10
27 alcom 1737 . . . . . . . . . 10
2826, 27anbi12i 678 . . . . . . . . 9
2924, 28bitri 240 . . . . . . . 8
3029albii 1566 . . . . . . 7
3123, 30bitr4i 243 . . . . . 6
32 19.23v 1891 . . . . . . . 8
33 19.23v 1891 . . . . . . . 8
3432, 33anbi12i 678 . . . . . . 7
35342albii 1567 . . . . . 6
36 nfe1 1732 . . . . . . . 8
37 nfv 1619 . . . . . . . 8
3836, 37nfim 1813 . . . . . . 7
39 nfe1 1732 . . . . . . . 8
40 nfv 1619 . . . . . . . 8
4139, 40nfim 1813 . . . . . . 7
4238, 41aaan 1884 . . . . . 6
4331, 35, 423bitri 262 . . . . 5
44432exbii 1583 . . . 4
45 eeanv 1913 . . . 4
4644, 45bitr2i 241 . . 3
4710, 46anbi12i 678 . 2
485, 6, 473bitri 262 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358  wal 1540  wex 1541   wceq 1642  weu 2204 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 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 This theorem is referenced by:  2eu5  2288  2eu6  2289
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