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Theorem nfrexxy 2505
Description: Not-free for restricted existential quantification where  x and  y are distinct. See nfrexya 2507 for a version with  y and 
A distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
Hypotheses
Ref Expression
nfralxy.1  |-  F/_ x A
nfralxy.2  |-  F/ x ph
Assertion
Ref Expression
nfrexxy  |-  F/ x E. y  e.  A  ph
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    A( x, y)

Proof of Theorem nfrexxy
StepHypRef Expression
1 nftru 1454 . . 3  |-  F/ y T.
2 nfralxy.1 . . . 4  |-  F/_ x A
32a1i 9 . . 3  |-  ( T. 
->  F/_ x A )
4 nfralxy.2 . . . 4  |-  F/ x ph
54a1i 9 . . 3  |-  ( T. 
->  F/ x ph )
61, 3, 5nfrexdxy 2500 . 2  |-  ( T. 
->  F/ x E. y  e.  A  ph )
76mptru 1352 1  |-  F/ x E. y  e.  A  ph
Colors of variables: wff set class
Syntax hints:   T. wtru 1344   F/wnf 1448   F/_wnfc 2295   E.wrex 2445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-i5r 1523  ax-ext 2147
This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450
This theorem is referenced by:  r19.12  2572  sbcrext  3028  nfuni  3795  nfiunxy  3892  rexxpf  4751  abrexex2g  6088  abrexex2  6092  nfrecs  6275  fimaxre2  11168  nfsum  11298  nfcprod1  11495  nfcprod  11496  bezoutlemmain  11931  ctiunctlemfo  12372  bj-findis  13861  strcollnfALT  13868
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