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Theorem nfrexxy 2361
Description: Not-free for restricted existential quantification where 𝑥 and 𝑦 are distinct. See nfrexya 2363 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 𝑥𝑦𝐴 𝜑
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfrexxy
StepHypRef Expression
1 nftru 1355 . . 3 𝑦
2 nfralxy.1 . . . 4 𝑥𝐴
32a1i 9 . . 3 (⊤ → 𝑥𝐴)
4 nfralxy.2 . . . 4 𝑥𝜑
54a1i 9 . . 3 (⊤ → Ⅎ𝑥𝜑)
61, 3, 5nfrexdxy 2357 . 2 (⊤ → Ⅎ𝑥𝑦𝐴 𝜑)
76trud 1252 1 𝑥𝑦𝐴 𝜑
Colors of variables: wff set class
Syntax hints:  wtru 1244  wnf 1349  wnfc 2165  wrex 2307
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 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312
This theorem is referenced by:  r19.12  2422  sbcrext  2835  nfuni  3586  nfiunxy  3683  rexxpf  4483  abrexex2g  5747  abrexex2  5751  nfrecs  5922  bj-findis  10078  strcollnfALT  10085
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