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Theorem nfrexxy 2475
Description: Not-free for restricted existential quantification where 𝑥 and 𝑦 are distinct. See nfrexya 2477 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 1443 . . 3 𝑦
2 nfralxy.1 . . . 4 𝑥𝐴
32a1i 9 . . 3 (⊤ → 𝑥𝐴)
4 nfralxy.2 . . . 4 𝑥𝜑
54a1i 9 . . 3 (⊤ → Ⅎ𝑥𝜑)
61, 3, 5nfrexdxy 2471 . 2 (⊤ → Ⅎ𝑥𝑦𝐴 𝜑)
76mptru 1341 1 𝑥𝑦𝐴 𝜑
Colors of variables: wff set class
Syntax hints:  wtru 1333  wnf 1437  wnfc 2269  wrex 2418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423
This theorem is referenced by:  r19.12  2541  sbcrext  2989  nfuni  3749  nfiunxy  3846  rexxpf  4693  abrexex2g  6025  abrexex2  6029  nfrecs  6211  fimaxre2  11029  nfsum  11157  nfcprod1  11354  nfcprod  11355  bezoutlemmain  11720  ctiunctlemfo  11986  bj-findis  13346  strcollnfALT  13353
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