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Theorem nfreudxy 2540
Description: Not-free deduction for restricted uniqueness. This is a version where  x and  y are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.)
Hypotheses
Ref Expression
nfreudxy.1  |-  F/ y
ph
nfreudxy.2  |-  ( ph  -> 
F/_ x A )
nfreudxy.3  |-  ( ph  ->  F/ x ps )
Assertion
Ref Expression
nfreudxy  |-  ( ph  ->  F/ x E! y  e.  A  ps )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)    A( x, y)

Proof of Theorem nfreudxy
StepHypRef Expression
1 nfreudxy.1 . . 3  |-  F/ y
ph
2 nfcv 2228 . . . . . 6  |-  F/_ x
y
32a1i 9 . . . . 5  |-  ( ph  -> 
F/_ x y )
4 nfreudxy.2 . . . . 5  |-  ( ph  -> 
F/_ x A )
53, 4nfeld 2244 . . . 4  |-  ( ph  ->  F/ x  y  e.  A )
6 nfreudxy.3 . . . 4  |-  ( ph  ->  F/ x ps )
75, 6nfand 1505 . . 3  |-  ( ph  ->  F/ x ( y  e.  A  /\  ps ) )
81, 7nfeud 1964 . 2  |-  ( ph  ->  F/ x E! y ( y  e.  A  /\  ps ) )
9 df-reu 2366 . . 3  |-  ( E! y  e.  A  ps  <->  E! y ( y  e.  A  /\  ps )
)
109nfbii 1407 . 2  |-  ( F/ x E! y  e.  A  ps  <->  F/ x E! y ( y  e.  A  /\  ps )
)
118, 10sylibr 132 1  |-  ( ph  ->  F/ x E! y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   F/wnf 1394    e. wcel 1438   E!weu 1948   F/_wnfc 2215   E!wreu 2361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-cleq 2081  df-clel 2084  df-nfc 2217  df-reu 2366
This theorem is referenced by:  nfreuxy  2541
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