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Theorem 2eu5 2227
Description: An alternate definition of double existential uniqueness (see 2eu4 2226). 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 2223 . . 3  |-  ( A. x E* y ph  ->  ( E! x E! y
ph 
<->  ( E! x E. y ph  /\  E! y E. x ph )
) )
21pm5.32ri 619 . 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 2183 . . . . 5  |-  ( E! y E. x ph  ->  E* y E. x ph )
43adantl 452 . . . 4  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  ->  E* y E. x ph )
5 2moex 2214 . . . 4  |-  ( E* y E. x ph  ->  A. x E* y ph )
64, 5syl 15 . . 3  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  ->  A. x E* y ph )
76pm4.71i 613 . 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 2226 . 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 264 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528   E!weu 2143   E*wmo 2144
This theorem is referenced by:  2reu5lem3  2972
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148
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