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Theorem euxfr 3120
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
euxfr.1  |-  A  e. 
_V
euxfr.2  |-  E! y  x  =  A
euxfr.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
euxfr  |-  ( 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 euxfr
StepHypRef Expression
1 euxfr.2 . . . . . 6  |-  E! y  x  =  A
2 euex 2304 . . . . . 6  |-  ( E! y  x  =  A  ->  E. y  x  =  A )
31, 2ax-mp 8 . . . . 5  |-  E. y  x  =  A
43biantrur 493 . . . 4  |-  ( ph  <->  ( E. y  x  =  A  /\  ph )
)
5 19.41v 1924 . . . 4  |-  ( E. y ( x  =  A  /\  ph )  <->  ( E. y  x  =  A  /\  ph )
)
6 euxfr.3 . . . . . 6  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
76pm5.32i 619 . . . . 5  |-  ( ( x  =  A  /\  ph )  <->  ( x  =  A  /\  ps )
)
87exbii 1592 . . . 4  |-  ( E. y ( x  =  A  /\  ph )  <->  E. y ( x  =  A  /\  ps )
)
94, 5, 83bitr2i 265 . . 3  |-  ( ph  <->  E. y ( x  =  A  /\  ps )
)
109eubii 2290 . 2  |-  ( E! x ph  <->  E! x E. y ( x  =  A  /\  ps )
)
11 euxfr.1 . . 3  |-  A  e. 
_V
121eumoi 2322 . . 3  |-  E* y  x  =  A
1311, 12euxfr2 3119 . 2  |-  ( E! x E. y ( x  =  A  /\  ps )  <->  E! y ps )
1410, 13bitri 241 1  |-  ( E! x ph  <->  E! y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   E!weu 2281   _Vcvv 2956
This theorem is referenced by:  moxfr  26733
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-v 2958
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