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Theorem euxfr 3289
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 2303 . . . . . 6  |-  ( E! y  x  =  A  ->  E. y  x  =  A )
31, 2ax-mp 5 . . . . 5  |-  E. y  x  =  A
43biantrur 506 . . . 4  |-  ( ph  <->  ( E. y  x  =  A  /\  ph )
)
5 19.41v 1945 . . . 4  |-  ( E. y ( x  =  A  /\  ph )  <->  ( E. y  x  =  A  /\  ph )
)
6 euxfr.3 . . . . . 6  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
76pm5.32i 637 . . . . 5  |-  ( ( x  =  A  /\  ph )  <->  ( x  =  A  /\  ps )
)
87exbii 1644 . . . 4  |-  ( E. y ( x  =  A  /\  ph )  <->  E. y ( x  =  A  /\  ps )
)
94, 5, 83bitr2i 273 . . 3  |-  ( ph  <->  E. y ( x  =  A  /\  ps )
)
109eubii 2300 . 2  |-  ( E! x ph  <->  E! x E. y ( x  =  A  /\  ps )
)
11 euxfr.1 . . 3  |-  A  e. 
_V
121eumoi 2309 . . 3  |-  E* y  x  =  A
1311, 12euxfr2 3288 . 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 369    = wceq 1379   E.wex 1596    e. wcel 1767   E!weu 2275   _Vcvv 3113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-v 3115
This theorem is referenced by:  moxfr  30228
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