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Theorem euxfr 2985
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 2199 . . . . . 6  |-  ( E! y  x  =  A  ->  E. y  x  =  A )
31, 2ax-mp 8 . . . . 5  |-  E. y  x  =  A
43biantrur 492 . . . 4  |-  ( ph  <->  ( E. y  x  =  A  /\  ph )
)
5 19.41v 1873 . . . 4  |-  ( E. y ( x  =  A  /\  ph )  <->  ( E. y  x  =  A  /\  ph )
)
6 euxfr.3 . . . . . 6  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
76pm5.32i 618 . . . . 5  |-  ( ( x  =  A  /\  ph )  <->  ( x  =  A  /\  ps )
)
87exbii 1573 . . . 4  |-  ( E. y ( x  =  A  /\  ph )  <->  E. y ( x  =  A  /\  ps )
)
94, 5, 83bitr2i 264 . . 3  |-  ( ph  <->  E. y ( x  =  A  /\  ps )
)
109eubii 2185 . 2  |-  ( E! x ph  <->  E! x E. y ( x  =  A  /\  ps )
)
11 euxfr.1 . . 3  |-  A  e. 
_V
121eumoi 2217 . . 3  |-  E* y  x  =  A
1311, 12euxfr2 2984 . 2  |-  ( E! x E. y ( x  =  A  /\  ps )  <->  E! y ps )
1410, 13bitri 240 1  |-  ( E! x ph  <->  E! y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1532    = wceq 1633    e. wcel 1701   E!weu 2176   _Vcvv 2822
This theorem is referenced by:  moxfr  25900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-v 2824
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