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Theorem 2eu4 2380
 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 2376 for a condition under which the naive definition holds and 2exeu 2371 for a one-way implication. See 2eu5 2382 and 2eu8 2386 for alternate definitions. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 14-Sep-2019.)
Assertion
Ref Expression
2eu4
Distinct variable groups:   ,,,   ,,

Proof of Theorem 2eu4
StepHypRef Expression
1 eu5 2310 . . 3
2 eu5 2310 . . . 4
3 excom 1849 . . . . 5
43anbi1i 695 . . . 4
52, 4bitri 249 . . 3
61, 5anbi12i 697 . 2
7 anandi 828 . 2
8 2mo2 2372 . . 3
98anbi2i 694 . 2
106, 7, 93bitr2i 273 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612  E!weu 2282  E*wmo 2283 This theorem is referenced by:  2eu5  2382  2eu6  2383  2eu6OLD  2384 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854 This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-nf 1617  df-eu 2286  df-mo 2287
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