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Theorem euxfr2 3245
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
euxfr2.1  |-  A  e. 
_V
euxfr2.2  |-  E* y  x  =  A
Assertion
Ref Expression
euxfr2  |-  ( E! x E. y ( x  =  A  /\  ph )  <->  E! y ph )
Distinct variable groups:    ph, x    x, A
Allowed substitution hints:    ph( y)    A( y)

Proof of Theorem euxfr2
StepHypRef Expression
1 2euswap 2365 . . . 4  |-  ( A. x E* y ( x  =  A  /\  ph )  ->  ( E! x E. y ( x  =  A  /\  ph )  ->  E! y E. x
( x  =  A  /\  ph ) ) )
2 euxfr2.2 . . . . . 6  |-  E* y  x  =  A
32moani 2334 . . . . 5  |-  E* y
( ph  /\  x  =  A )
4 ancom 450 . . . . . 6  |-  ( (
ph  /\  x  =  A )  <->  ( x  =  A  /\  ph )
)
54mobii 2287 . . . . 5  |-  ( E* y ( ph  /\  x  =  A )  <->  E* y ( x  =  A  /\  ph )
)
63, 5mpbi 208 . . . 4  |-  E* y
( x  =  A  /\  ph )
71, 6mpg 1594 . . 3  |-  ( E! x E. y ( x  =  A  /\  ph )  ->  E! y E. x ( x  =  A  /\  ph )
)
8 2euswap 2365 . . . 4  |-  ( A. y E* x ( x  =  A  /\  ph )  ->  ( E! y E. x ( x  =  A  /\  ph )  ->  E! x E. y ( x  =  A  /\  ph )
) )
9 moeq 3236 . . . . . 6  |-  E* x  x  =  A
109moani 2334 . . . . 5  |-  E* x
( ph  /\  x  =  A )
114mobii 2287 . . . . 5  |-  ( E* x ( ph  /\  x  =  A )  <->  E* x ( x  =  A  /\  ph )
)
1210, 11mpbi 208 . . . 4  |-  E* x
( x  =  A  /\  ph )
138, 12mpg 1594 . . 3  |-  ( E! y E. x ( x  =  A  /\  ph )  ->  E! x E. y ( x  =  A  /\  ph )
)
147, 13impbii 188 . 2  |-  ( E! x E. y ( x  =  A  /\  ph )  <->  E! y E. x
( x  =  A  /\  ph ) )
15 euxfr2.1 . . . 4  |-  A  e. 
_V
16 biidd 237 . . . 4  |-  ( x  =  A  ->  ( ph 
<-> 
ph ) )
1715, 16ceqsexv 3109 . . 3  |-  ( E. x ( x  =  A  /\  ph )  <->  ph )
1817eubii 2286 . 2  |-  ( E! y E. x ( x  =  A  /\  ph )  <->  E! y ph )
1914, 18bitri 249 1  |-  ( E! x E. y ( x  =  A  /\  ph )  <->  E! y ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758   E!weu 2261   E*wmo 2262   _Vcvv 3072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-v 3074
This theorem is referenced by:  euxfr  3246  euop2  4694
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