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Theorem nfrexxy 2355
Description: Not-free for restricted existential quantification where x and y are distinct. See nfrexya 2357 for a version with y and A distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
Hypotheses
Ref Expression
nfralxy.1 xA
nfralxy.2 xφ
Assertion
Ref Expression
nfrexxy xy A φ
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)   A(x,y)

Proof of Theorem nfrexxy
StepHypRef Expression
1 nftru 1352 . . 3 y
2 nfralxy.1 . . . 4 xA
32a1i 9 . . 3 ( ⊤ → xA)
4 nfralxy.2 . . . 4 xφ
54a1i 9 . . 3 ( ⊤ → Ⅎxφ)
61, 3, 5nfrexdxy 2351 . 2 ( ⊤ → Ⅎxy A φ)
76trud 1251 1 xy A φ
Colors of variables: wff set class
Syntax hints:  wtru 1243  wnf 1346  wnfc 2162  wrex 2301
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306
This theorem is referenced by:  r19.12  2416  sbcrext  2829  sbcrexg  2831  nfuni  3577  nfiunxy  3674  rexxpf  4426  abrexex2g  5689  abrexex2  5693  nfrecs  5863  bj-findis  9409  strcollnfALT  9416
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