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Theorem reu4 2799
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 2574 . 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 2798 . . 3  |-  ( E* x  e.  A  ph  <->  A. x  e.  A  A. y  e.  A  (
( ph  /\  ps )  ->  x  =  y ) )
43anbi2i 445 . 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 182 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 102    <-> wb 103   A.wral 2355   E.wrex 2356   E!wreu 2357   E*wrmo 2358
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-cleq 2078  df-clel 2081  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363
This theorem is referenced by:  reuind  2808  receuap  8054  lbreu  8318  cju  8333  ndvdssub  10724  qredeu  10873
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