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Theorem hbeu1 1952
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 1945 . 2  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
2 hba1 1474 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  A. x A. x ( ph  <->  x  =  y ) )
32hbex 1568 . 2  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  A. x E. y A. x ( ph  <->  x  =  y ) )
41, 3hbxfrbi 1402 1  |-  ( E! x ph  ->  A. x E! x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283   E.wex 1422   E!weu 1942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-eu 1945
This theorem is referenced by:  hbmo1  1980  eupicka  2022  exists2  2039
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