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Theorem nfrexxy 2426
Description: Not-free for restricted existential quantification where 𝑥 and 𝑦 are distinct. See nfrexya 2428 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 1407 . . 3 𝑦
2 nfralxy.1 . . . 4 𝑥𝐴
32a1i 9 . . 3 (⊤ → 𝑥𝐴)
4 nfralxy.2 . . . 4 𝑥𝜑
54a1i 9 . . 3 (⊤ → Ⅎ𝑥𝜑)
61, 3, 5nfrexdxy 2422 . 2 (⊤ → Ⅎ𝑥𝑦𝐴 𝜑)
76mptru 1305 1 𝑥𝑦𝐴 𝜑
Colors of variables: wff set class
Syntax hints:  wtru 1297  wnf 1401  wnfc 2222  wrex 2371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-4 1452  ax-17 1471  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rex 2376
This theorem is referenced by:  r19.12  2491  sbcrext  2930  nfuni  3681  nfiunxy  3778  rexxpf  4614  abrexex2g  5929  abrexex2  5933  nfrecs  6110  fimaxre2  10773  nfsum  10900  bezoutlemmain  11414  bj-findis  12582  strcollnfALT  12589
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