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Theorem nfrexxy 2378
Description: Not-free for restricted existential quantification where  x and  y are distinct. See nfrexya 2380 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 1371 . . 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 2374 . 2  |-  ( T. 
->  F/ x E. y  e.  A  ph )
76trud 1268 1  |-  F/ x E. y  e.  A  ph
Colors of variables: wff set class
Syntax hints:   T. wtru 1260   F/wnf 1365   F/_wnfc 2181   E.wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329
This theorem is referenced by:  r19.12  2439  sbcrext  2863  nfuni  3614  nfiunxy  3711  rexxpf  4511  abrexex2g  5775  abrexex2  5779  nfrecs  5953  bj-findis  10491  strcollnfALT  10498
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