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Theorem 2eu5 2406
Description: An alternate definition of double existential uniqueness (see 2eu4 2405). A mistake sometimes made in the literature is to use  E! x E! y to mean "exactly one  x and exactly one  y." (For example, see Proposition 7.53 of [TakeutiZaring] p. 53.) It turns out that this is actually a weaker assertion, as can be seen by expanding out the formal definitions. This theorem shows that the erroneous definition can be repaired by conjoining 
A. x E* y ph as an additional condition. The correct definition apparently has never been published. ( E* means "exists at most one."). (Contributed by NM, 26-Oct-2003.)
Assertion
Ref Expression
2eu5  |-  ( ( E! x E! y
ph  /\  A. x E* y ph )  <->  ( E. x E. y ph  /\  E. z E. w A. x A. y ( ph  ->  ( x  =  z  /\  y  =  w ) ) ) )
Distinct variable groups:    x, y,
z, w    ph, z, w
Allowed substitution hints:    ph( x, y)

Proof of Theorem 2eu5
StepHypRef Expression
1 2eu1 2402 . . 3  |-  ( A. x E* y ph  ->  ( E! x E! y
ph 
<->  ( E! x E. y ph  /\  E! y E. x ph )
) )
21pm5.32ri 650 . 2  |-  ( ( E! x E! y
ph  /\  A. x E* y ph )  <->  ( ( E! x E. y ph  /\  E! y E. x ph )  /\  A. x E* y ph ) )
3 eumo 2348 . . . . 5  |-  ( E! y E. x ph  ->  E* y E. x ph )
43adantl 473 . . . 4  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  ->  E* y E. x ph )
5 2moex 2392 . . . 4  |-  ( E* y E. x ph  ->  A. x E* y ph )
64, 5syl 17 . . 3  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  ->  A. x E* y ph )
76pm4.71i 644 . 2  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  ( ( E! x E. y ph  /\  E! y E. x ph )  /\  A. x E* y ph ) )
8 2eu4 2405 . 2  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  ( E. x E. y ph  /\  E. z E. w A. x A. y ( ph  ->  ( x  =  z  /\  y  =  w ) ) ) )
92, 7, 83bitr2i 281 1  |-  ( ( E! x E! y
ph  /\  A. x E* y ph )  <->  ( E. x E. y ph  /\  E. z E. w A. x A. y ( ph  ->  ( x  =  z  /\  y  =  w ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376   A.wal 1450   E.wex 1671   E!weu 2319   E*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  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-eu 2323  df-mo 2324
This theorem is referenced by:  2reu5lem3  3235
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