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Theorem nfrexxy 2529
Description: Not-free for restricted existential quantification where 𝑥 and 𝑦 are distinct. See nfrexya 2531 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 1477 . . 3 𝑦
2 nfralxy.1 . . . 4 𝑥𝐴
32a1i 9 . . 3 (⊤ → 𝑥𝐴)
4 nfralxy.2 . . . 4 𝑥𝜑
54a1i 9 . . 3 (⊤ → Ⅎ𝑥𝜑)
61, 3, 5nfrexdxy 2524 . 2 (⊤ → Ⅎ𝑥𝑦𝐴 𝜑)
76mptru 1373 1 𝑥𝑦𝐴 𝜑
Colors of variables: wff set class
Syntax hints:  wtru 1365  wnf 1471  wnfc 2319  wrex 2469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474
This theorem is referenced by:  r19.12  2596  sbcrext  3055  nfuni  3830  nfiunxy  3927  rexxpf  4792  abrexex2g  6145  abrexex2  6149  nfrecs  6332  fimaxre2  11267  nfsum  11397  nfcprod1  11594  nfcprod  11595  bezoutlemmain  12031  ctiunctlemfo  12490  bj-findis  15189  strcollnfALT  15196
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