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Theorem nfrexxy 2415
Description: Not-free for restricted existential quantification where  x and  y are distinct. See nfrexya 2417 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 1400 . . 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 2411 . 2  |-  ( T. 
->  F/ x E. y  e.  A  ph )
76mptru 1298 1  |-  F/ x E. y  e.  A  ph
Colors of variables: wff set class
Syntax hints:   T. wtru 1290   F/wnf 1394   F/_wnfc 2215   E.wrex 2360
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-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365
This theorem is referenced by:  r19.12  2478  sbcrext  2914  nfuni  3654  nfiunxy  3751  rexxpf  4571  abrexex2g  5873  abrexex2  5877  nfrecs  6054  fimaxre2  10622  nfsum  10710  bezoutlemmain  11080  bj-findis  11531  strcollnfALT  11538
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