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Theorem reu4 2873
Description: Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.)
Hypothesis
Ref Expression
rmo4.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
reu4  |-  ( E! x  e.  A  ph  <->  ( E. x  e.  A  ph 
/\  A. x  e.  A  A. y  e.  A  ( ( ph  /\  ps )  ->  x  =  y ) ) )
Distinct variable groups:    x, y, A    ph, y    ps, x
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem reu4
StepHypRef Expression
1 reu5 2641 . 2  |-  ( E! x  e.  A  ph  <->  ( E. x  e.  A  ph 
/\  E* x  e.  A  ph ) )
2 rmo4.1 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
32rmo4 2872 . . 3  |-  ( E* x  e.  A  ph  <->  A. x  e.  A  A. y  e.  A  (
( ph  /\  ps )  ->  x  =  y ) )
43anbi2i 452 . 2  |-  ( ( E. x  e.  A  ph 
/\  E* x  e.  A  ph )  <->  ( E. x  e.  A  ph  /\  A. x  e.  A  A. y  e.  A  (
( ph  /\  ps )  ->  x  =  y ) ) )
51, 4bitri 183 1  |-  ( E! x  e.  A  ph  <->  ( E. x  e.  A  ph 
/\  A. x  e.  A  A. y  e.  A  ( ( ph  /\  ps )  ->  x  =  y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wral 2414   E.wrex 2415   E!wreu 2416   E*wrmo 2417
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-cleq 2130  df-clel 2133  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422
This theorem is referenced by:  reuind  2884  receuap  8423  lbreu  8696  cju  8712  ndvdssub  11616  qredeu  11767
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