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Theorem euxfr2 3288
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 2379 . . . 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 2348 . . . . 5  |-  E* y
( ph  /\  x  =  A )
4 ancom 450 . . . . . 6  |-  ( (
ph  /\  x  =  A )  <->  ( x  =  A  /\  ph )
)
54mobii 2301 . . . . 5  |-  ( E* y ( ph  /\  x  =  A )  <->  E* y ( x  =  A  /\  ph )
)
63, 5mpbi 208 . . . 4  |-  E* y
( x  =  A  /\  ph )
71, 6mpg 1603 . . 3  |-  ( E! x E. y ( x  =  A  /\  ph )  ->  E! y E. x ( x  =  A  /\  ph )
)
8 2euswap 2379 . . . 4  |-  ( A. y E* x ( x  =  A  /\  ph )  ->  ( E! y E. x ( x  =  A  /\  ph )  ->  E! x E. y ( x  =  A  /\  ph )
) )
9 moeq 3279 . . . . . 6  |-  E* x  x  =  A
109moani 2348 . . . . 5  |-  E* x
( ph  /\  x  =  A )
114mobii 2301 . . . . 5  |-  ( E* x ( ph  /\  x  =  A )  <->  E* x ( x  =  A  /\  ph )
)
1210, 11mpbi 208 . . . 4  |-  E* x
( x  =  A  /\  ph )
138, 12mpg 1603 . . 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 3150 . . 3  |-  ( E. x ( x  =  A  /\  ph )  <->  ph )
1817eubii 2300 . 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 1379   E.wex 1596    e. wcel 1767   E!weu 2275   E*wmo 2276   _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:  euxfr  3289  euop2  4747
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