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Theorem eumo0 2006
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 1980 . 2  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
3 bi1 117 . . . 4  |-  ( (
ph 
<->  x  =  y )  ->  ( ph  ->  x  =  y ) )
43alimi 1414 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  A. x
( ph  ->  x  =  y ) )
54eximi 1562 . 2  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  E. y A. x
( ph  ->  x  =  y ) )
62, 5sylbi 120 1  |-  ( E! x ph  ->  E. y A. x ( ph  ->  x  =  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1312   E.wex 1451   E!weu 1975
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-eu 1978
This theorem is referenced by:  eu2  2019  eu3h  2020
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