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Theorem euxfr2 3055
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 2307 . . . 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 2283 . . . . 5  |-  E* y
( ph  /\  x  =  A )
4 ancom 438 . . . . . 6  |-  ( (
ph  /\  x  =  A )  <->  ( x  =  A  /\  ph )
)
54mobii 2267 . . . . 5  |-  ( E* y ( ph  /\  x  =  A )  <->  E* y ( x  =  A  /\  ph )
)
63, 5mpbi 200 . . . 4  |-  E* y
( x  =  A  /\  ph )
71, 6mpg 1554 . . 3  |-  ( E! x E. y ( x  =  A  /\  ph )  ->  E! y E. x ( x  =  A  /\  ph )
)
8 2euswap 2307 . . . 4  |-  ( A. y E* x ( x  =  A  /\  ph )  ->  ( E! y E. x ( x  =  A  /\  ph )  ->  E! x E. y ( x  =  A  /\  ph )
) )
9 moeq 3046 . . . . . 6  |-  E* x  x  =  A
109moani 2283 . . . . 5  |-  E* x
( ph  /\  x  =  A )
114mobii 2267 . . . . 5  |-  ( E* x ( ph  /\  x  =  A )  <->  E* x ( x  =  A  /\  ph )
)
1210, 11mpbi 200 . . . 4  |-  E* x
( x  =  A  /\  ph )
138, 12mpg 1554 . . 3  |-  ( E! y E. x ( x  =  A  /\  ph )  ->  E! x E. y ( x  =  A  /\  ph )
)
147, 13impbii 181 . 2  |-  ( E! x E. y ( x  =  A  /\  ph )  <->  E! y E. x
( x  =  A  /\  ph ) )
15 euxfr2.1 . . . 4  |-  A  e. 
_V
16 biidd 229 . . . 4  |-  ( x  =  A  ->  ( ph 
<-> 
ph ) )
1715, 16ceqsexv 2927 . . 3  |-  ( E. x ( x  =  A  /\  ph )  <->  ph )
1817eubii 2240 . 2  |-  ( E! y E. x ( x  =  A  /\  ph )  <->  E! y ph )
1914, 18bitri 241 1  |-  ( E! x E. y ( x  =  A  /\  ph )  <->  E! y ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   E!weu 2231   E*wmo 2232   _Vcvv 2892
This theorem is referenced by:  euxfr  3056  euop2  4390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-v 2894
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