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Theorem euxfr 3210
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 2244 . . . . . 6  |-  ( E! y  x  =  A  ->  E. y  x  =  A )
31, 2ax-mp 5 . . . . 5  |-  E. y  x  =  A
43biantrur 504 . . . 4  |-  ( ph  <->  ( E. y  x  =  A  /\  ph )
)
5 19.41v 1779 . . . 4  |-  ( E. y ( x  =  A  /\  ph )  <->  ( E. y  x  =  A  /\  ph )
)
6 euxfr.3 . . . . . 6  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
76pm5.32i 635 . . . . 5  |-  ( ( x  =  A  /\  ph )  <->  ( x  =  A  /\  ps )
)
87exbii 1675 . . . 4  |-  ( E. y ( x  =  A  /\  ph )  <->  E. y ( x  =  A  /\  ps )
)
94, 5, 83bitr2i 273 . . 3  |-  ( ph  <->  E. y ( x  =  A  /\  ps )
)
109eubii 2242 . 2  |-  ( E! x ph  <->  E! x E. y ( x  =  A  /\  ps )
)
11 euxfr.1 . . 3  |-  A  e. 
_V
121eumoi 2250 . . 3  |-  E* y  x  =  A
1311, 12euxfr2 3209 . 2  |-  ( E! x E. y ( x  =  A  /\  ps )  <->  E! y ps )
1410, 13bitri 249 1  |-  ( E! x ph  <->  E! y ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399   E.wex 1620    e. wcel 1826   E!weu 2218   _Vcvv 3034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-v 3036
This theorem is referenced by:  moxfr  30790
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