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Theorem hbeu1 1985
Description: Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.)
Assertion
Ref Expression
hbeu1  |-  ( E! x ph  ->  A. x E! x ph )

Proof of Theorem hbeu1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-eu 1978 . 2  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
2 hba1 1503 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  A. x A. x ( ph  <->  x  =  y ) )
32hbex 1598 . 2  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  A. x E. y A. x ( ph  <->  x  =  y ) )
41, 3hbxfrbi 1431 1  |-  ( E! x ph  ->  A. x E! x ph )
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-mp 5  ax-1 6  ax-2 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-4 1470  ax-ial 1497
This theorem depends on definitions:  df-bi 116  df-eu 1978
This theorem is referenced by:  hbmo1  2013  eupicka  2055  exists2  2072
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