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Theorem euxfr2dc 2934
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 2106 . . . . . 6  |-  E* y
( ph  /\  x  =  A )
3 ancom 266 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  <->  ( x  =  A  /\  ph )
)
43mobii 2073 . . . . . 6  |-  ( E* y ( ph  /\  x  =  A )  <->  E* y ( x  =  A  /\  ph )
)
52, 4mpbi 145 . . . . 5  |-  E* y
( x  =  A  /\  ph )
65ax-gen 1459 . . . 4  |-  A. x E* y ( x  =  A  /\  ph )
7 excom 1674 . . . . . 6  |-  ( E. y E. x ( x  =  A  /\  ph )  <->  E. x E. y
( x  =  A  /\  ph ) )
87dcbii 841 . . . . 5  |-  (DECID  E. y E. x ( x  =  A  /\  ph )  <-> DECID  E. x E. y ( x  =  A  /\  ph )
)
9 2euswapdc 2127 . . . . 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 121 . . . 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 2924 . . . . . . 7  |-  E* x  x  =  A
1312moani 2106 . . . . . 6  |-  E* x
( ph  /\  x  =  A )
143mobii 2073 . . . . . 6  |-  ( E* x ( ph  /\  x  =  A )  <->  E* x ( x  =  A  /\  ph )
)
1513, 14mpbi 145 . . . . 5  |-  E* x
( x  =  A  /\  ph )
1615ax-gen 1459 . . . 4  |-  A. y E* x ( x  =  A  /\  ph )
17 2euswapdc 2127 . . . 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 129 . 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 172 . . . 4  |-  ( x  =  A  ->  ( ph 
<-> 
ph ) )
2220, 21ceqsexv 2788 . . 3  |-  ( E. x ( x  =  A  /\  ph )  <->  ph )
2322eubii 2045 . 2  |-  ( E! y E. x ( x  =  A  /\  ph )  <->  E! y ph )
2419, 23bitrdi 196 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 104    <-> wb 105  DECID wdc 835   A.wal 1361    = wceq 1363   E.wex 1502   E!weu 2036   E*wmo 2037    e. wcel 2158   _Vcvv 2749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-v 2751
This theorem is referenced by:  euxfrdc  2935
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