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Theorem nfralxy 2448
Description: Not-free for restricted universal quantification where 𝑥 and 𝑦 are distinct. See nfralya 2450 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
nfralxy 𝑥𝑦𝐴 𝜑
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfralxy
StepHypRef Expression
1 nftru 1427 . . 3 𝑦
2 nfralxy.1 . . . 4 𝑥𝐴
32a1i 9 . . 3 (⊤ → 𝑥𝐴)
4 nfralxy.2 . . . 4 𝑥𝜑
54a1i 9 . . 3 (⊤ → Ⅎ𝑥𝜑)
61, 3, 5nfraldxy 2444 . 2 (⊤ → Ⅎ𝑥𝑦𝐴 𝜑)
76mptru 1325 1 𝑥𝑦𝐴 𝜑
Colors of variables: wff set class
Syntax hints:  wtru 1317  wnf 1421  wnfc 2245  wral 2393
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 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398
This theorem is referenced by:  nfra2xy  2452  rspc2  2774  sbcralt  2957  sbcralg  2959  raaanlem  3438  nfint  3751  nfiinxy  3810  nfpo  4193  nfso  4194  nfse  4233  nffrfor  4240  nfwe  4247  ralxpf  4655  funimaexglem  5176  fun11iun  5356  dff13f  5639  nfiso  5675  mpoeq123  5798  nfofr  5956  fmpox  6066  nfrecs  6172  xpf1o  6706  ac6sfi  6760  ismkvnex  6997  lble  8673  fzrevral  9853  nfsum1  11093  nfsum  11094  fsum2dlemstep  11171  fisumcom2  11175  bezoutlemmain  11613  cnmpt21  12387  setindis  13092  bdsetindis  13094  isomninnlem  13152
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