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Theorem eumo0 2069
Description: Existential uniqueness implies "at most one". (Contributed by NM, 8-Jul-1994.)
Hypothesis
Ref Expression
eumo0.1  |-  ( ph  ->  A. y ph )
Assertion
Ref Expression
eumo0  |-  ( E! x ph  ->  E. y A. x ( ph  ->  x  =  y ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem eumo0
StepHypRef Expression
1 eumo0.1 . . 3  |-  ( ph  ->  A. y ph )
21euf 2043 . 2  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
3 biimp 118 . . . 4  |-  ( (
ph 
<->  x  =  y )  ->  ( ph  ->  x  =  y ) )
43alimi 1466 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  A. x
( ph  ->  x  =  y ) )
54eximi 1611 . 2  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  E. y A. x
( ph  ->  x  =  y ) )
62, 5sylbi 121 1  |-  ( E! x ph  ->  E. y A. x ( ph  ->  x  =  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   A.wal 1362   E.wex 1503   E!weu 2038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546
This theorem depends on definitions:  df-bi 117  df-eu 2041
This theorem is referenced by:  eu2  2082  eu3h  2083
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