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Theorem nfreudxy 2661
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 2329 . . . . . 6  |-  F/_ x
y
32a1i 9 . . . . 5  |-  ( ph  -> 
F/_ x y )
4 nfreudxy.2 . . . . 5  |-  ( ph  -> 
F/_ x A )
53, 4nfeld 2345 . . . 4  |-  ( ph  ->  F/ x  y  e.  A )
6 nfreudxy.3 . . . 4  |-  ( ph  ->  F/ x ps )
75, 6nfand 1578 . . 3  |-  ( ph  ->  F/ x ( y  e.  A  /\  ps ) )
81, 7nfeud 2052 . 2  |-  ( ph  ->  F/ x E! y ( y  e.  A  /\  ps ) )
9 df-reu 2472 . . 3  |-  ( E! y  e.  A  ps  <->  E! y ( y  e.  A  /\  ps )
)
109nfbii 1483 . 2  |-  ( F/ x E! y  e.  A  ps  <->  F/ x E! y ( y  e.  A  /\  ps )
)
118, 10sylibr 134 1  |-  ( ph  ->  F/ x E! y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   F/wnf 1470   E!weu 2036    e. wcel 2158   F/_wnfc 2316   E!wreu 2467
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-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-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-cleq 2180  df-clel 2183  df-nfc 2318  df-reu 2472
This theorem is referenced by:  nfreuxy  2662
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