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Theorem eu4 2005
Description: Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)
Hypothesis
Ref Expression
eu4.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
eu4  |-  ( E! x ph  <->  ( E. x ph  /\  A. x A. y ( ( ph  /\ 
ps )  ->  x  =  y ) ) )
Distinct variable groups:    x, y    ph, y    ps, x
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem eu4
StepHypRef Expression
1 eu5 1990 . 2  |-  ( E! x ph  <->  ( E. x ph  /\  E* x ph ) )
2 eu4.1 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
32mo4 2004 . . 3  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
ps )  ->  x  =  y ) )
43anbi2i 445 . 2  |-  ( ( E. x ph  /\  E* x ph )  <->  ( E. x ph  /\  A. x A. y ( ( ph  /\ 
ps )  ->  x  =  y ) ) )
51, 4bitri 182 1  |-  ( E! x ph  <->  ( E. x ph  /\  A. x A. y ( ( ph  /\ 
ps )  ->  x  =  y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283   E.wex 1422   E!weu 1943   E*wmo 1944
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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947
This theorem is referenced by:  euequ1  2038  eueq  2772  euind  2788  eusv1  4230  eroveu  6284  climeu  10336
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