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Theorem euxfrdc 2779
Description: Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. (Contributed by NM, 14-Nov-2004.)
Hypotheses
Ref Expression
euxfrdc.1  |-  A  e. 
_V
euxfrdc.2  |-  E! y  x  =  A
euxfrdc.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
euxfrdc  |-  (DECID  E. y E. x ( x  =  A  /\  ps )  ->  ( E! x ph  <->  E! y ps ) )
Distinct variable groups:    ps, x    ph, y    x, A
Allowed substitution hints:    ph( x)    ps( y)    A( y)

Proof of Theorem euxfrdc
StepHypRef Expression
1 euxfrdc.2 . . . . . 6  |-  E! y  x  =  A
2 euex 1972 . . . . . 6  |-  ( E! y  x  =  A  ->  E. y  x  =  A )
31, 2ax-mp 7 . . . . 5  |-  E. y  x  =  A
43biantrur 297 . . . 4  |-  ( ph  <->  ( E. y  x  =  A  /\  ph )
)
5 19.41v 1824 . . . 4  |-  ( E. y ( x  =  A  /\  ph )  <->  ( E. y  x  =  A  /\  ph )
)
6 euxfrdc.3 . . . . . 6  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
76pm5.32i 442 . . . . 5  |-  ( ( x  =  A  /\  ph )  <->  ( x  =  A  /\  ps )
)
87exbii 1537 . . . 4  |-  ( E. y ( x  =  A  /\  ph )  <->  E. y ( x  =  A  /\  ps )
)
94, 5, 83bitr2i 206 . . 3  |-  ( ph  <->  E. y ( x  =  A  /\  ps )
)
109eubii 1951 . 2  |-  ( E! x ph  <->  E! x E. y ( x  =  A  /\  ps )
)
11 euxfrdc.1 . . 3  |-  A  e. 
_V
121eumoi 1975 . . 3  |-  E* y  x  =  A
1311, 12euxfr2dc 2778 . 2  |-  (DECID  E. y E. x ( x  =  A  /\  ps )  ->  ( E! x E. y ( x  =  A  /\  ps )  <->  E! y ps ) )
1410, 13syl5bb 190 1  |-  (DECID  E. y E. x ( x  =  A  /\  ps )  ->  ( E! x ph  <->  E! y ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103  DECID wdc 776    = wceq 1285   E.wex 1422    e. wcel 1434   E!weu 1942   _Vcvv 2602
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-in1 577  ax-in2 578  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  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-dc 777  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-v 2604
This theorem is referenced by: (None)
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