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Theorem reu4 2851
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 2620 . 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 2850 . . 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 2393   E.wrex 2394   E!wreu 2395   E*wrmo 2396
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-cleq 2110  df-clel 2113  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401
This theorem is referenced by:  reuind  2862  receuap  8398  lbreu  8671  cju  8687  ndvdssub  11554  qredeu  11705
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