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Theorem 2eu4 2405
 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 2402 for a condition under which the naive definition holds and 2exeu 2398 for a one-way implication. See 2eu5 2406 and 2eu8 2409 for alternate definitions. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 14-Sep-2019.)
Assertion
Ref Expression
2eu4
Distinct variable groups:   ,,,   ,,
Allowed substitution hints:   (,)

Proof of Theorem 2eu4
StepHypRef Expression
1 eu5 2345 . . 3
2 eu5 2345 . . . 4
3 excom 1944 . . . . 5
43anbi1i 709 . . . 4
52, 4bitri 257 . . 3
61, 5anbi12i 711 . 2
7 anandi 844 . 2
8 2mo2 2399 . . 3
98anbi2i 708 . 2
106, 7, 93bitr2i 281 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376  wal 1450  wex 1671  weu 2319  wmo 2320 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950 This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676  df-eu 2323  df-mo 2324 This theorem is referenced by:  2eu5  2406  2eu6  2407
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