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Theorem euxfr2dc 2749
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
euxfr2dc.1  |-  A  e. 
_V
euxfr2dc.2  |-  E* y  x  =  A
Assertion
Ref Expression
euxfr2dc  |-  (DECID  E. y E. x ( x  =  A  /\  ph )  ->  ( 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 euxfr2dc
StepHypRef Expression
1 euxfr2dc.2 . . . . . . 7  |-  E* y  x  =  A
21moani 1986 . . . . . 6  |-  E* y
( ph  /\  x  =  A )
3 ancom 257 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  <->  ( x  =  A  /\  ph )
)
43mobii 1953 . . . . . 6  |-  ( E* y ( ph  /\  x  =  A )  <->  E* y ( x  =  A  /\  ph )
)
52, 4mpbi 137 . . . . 5  |-  E* y
( x  =  A  /\  ph )
65ax-gen 1354 . . . 4  |-  A. x E* y ( x  =  A  /\  ph )
7 excom 1570 . . . . . 6  |-  ( E. y E. x ( x  =  A  /\  ph )  <->  E. x E. y
( x  =  A  /\  ph ) )
87dcbii 758 . . . . 5  |-  (DECID  E. y E. x ( x  =  A  /\  ph )  <-> DECID  E. x E. y ( x  =  A  /\  ph )
)
9 2euswapdc 2007 . . . . 5  |-  (DECID  E. x E. y ( x  =  A  /\  ph )  ->  ( A. x E* y ( x  =  A  /\  ph )  ->  ( E! x E. y ( x  =  A  /\  ph )  ->  E! y E. x
( x  =  A  /\  ph ) ) ) )
108, 9sylbi 118 . . . 4  |-  (DECID  E. y E. x ( x  =  A  /\  ph )  ->  ( A. x E* y ( x  =  A  /\  ph )  ->  ( E! x E. y ( x  =  A  /\  ph )  ->  E! y E. x
( x  =  A  /\  ph ) ) ) )
116, 10mpi 15 . . 3  |-  (DECID  E. y E. x ( x  =  A  /\  ph )  ->  ( E! x E. y ( x  =  A  /\  ph )  ->  E! y E. x
( x  =  A  /\  ph ) ) )
12 moeq 2739 . . . . . . 7  |-  E* x  x  =  A
1312moani 1986 . . . . . 6  |-  E* x
( ph  /\  x  =  A )
143mobii 1953 . . . . . 6  |-  ( E* x ( ph  /\  x  =  A )  <->  E* x ( x  =  A  /\  ph )
)
1513, 14mpbi 137 . . . . 5  |-  E* x
( x  =  A  /\  ph )
1615ax-gen 1354 . . . 4  |-  A. y E* x ( x  =  A  /\  ph )
17 2euswapdc 2007 . . . 4  |-  (DECID  E. y E. x ( x  =  A  /\  ph )  ->  ( A. y E* x ( x  =  A  /\  ph )  ->  ( E! y E. x ( x  =  A  /\  ph )  ->  E! x E. y
( x  =  A  /\  ph ) ) ) )
1816, 17mpi 15 . . 3  |-  (DECID  E. y E. x ( x  =  A  /\  ph )  ->  ( E! y E. x ( x  =  A  /\  ph )  ->  E! x E. y
( x  =  A  /\  ph ) ) )
1911, 18impbid 124 . 2  |-  (DECID  E. y E. x ( x  =  A  /\  ph )  ->  ( E! x E. y ( x  =  A  /\  ph )  <->  E! y E. x ( x  =  A  /\  ph ) ) )
20 euxfr2dc.1 . . . 4  |-  A  e. 
_V
21 biidd 165 . . . 4  |-  ( x  =  A  ->  ( ph 
<-> 
ph ) )
2220, 21ceqsexv 2610 . . 3  |-  ( E. x ( x  =  A  /\  ph )  <->  ph )
2322eubii 1925 . 2  |-  ( E! y E. x ( x  =  A  /\  ph )  <->  E! y ph )
2419, 23syl6bb 189 1  |-  (DECID  E. y E. x ( x  =  A  /\  ph )  ->  ( E! x E. y ( x  =  A  /\  ph )  <->  E! y ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102  DECID wdc 753   A.wal 1257    = wceq 1259   E.wex 1397    e. wcel 1409   E!weu 1916   E*wmo 1917   _Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-dc 754  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576
This theorem is referenced by:  euxfrdc  2750
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