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Theorem nfrexxy 2409
Description: Not-free for restricted existential quantification where 𝑥 and 𝑦 are distinct. See nfrexya 2411 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 1396 . . 3 𝑦
2 nfralxy.1 . . . 4 𝑥𝐴
32a1i 9 . . 3 (⊤ → 𝑥𝐴)
4 nfralxy.2 . . . 4 𝑥𝜑
54a1i 9 . . 3 (⊤ → Ⅎ𝑥𝜑)
61, 3, 5nfrexdxy 2405 . 2 (⊤ → Ⅎ𝑥𝑦𝐴 𝜑)
76trud 1294 1 𝑥𝑦𝐴 𝜑
Colors of variables: wff set class
Syntax hints:  wtru 1286  wnf 1390  wnfc 2210  wrex 2354
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359
This theorem is referenced by:  r19.12  2472  sbcrext  2902  nfuni  3633  nfiunxy  3730  rexxpf  4541  abrexex2g  5826  abrexex2  5830  nfrecs  6004  fimaxre2  10483  nfsum  10568  bezoutlemmain  10767  bj-findis  11217  strcollnfALT  11224
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