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Theorem nfralya 2530
Description: Not-free for restricted universal quantification where 𝑦 and 𝐴 are distinct. See nfralxy 2528 for a version with 𝑥 and 𝑦 distinct instead. (Contributed by Jim Kingdon, 3-Jun-2018.)
Hypotheses
Ref Expression
nfralya.1 𝑥𝐴
nfralya.2 𝑥𝜑
Assertion
Ref Expression
nfralya 𝑥𝑦𝐴 𝜑
Distinct variable group:   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem nfralya
StepHypRef Expression
1 nftru 1477 . . 3 𝑦
2 nfralya.1 . . . 4 𝑥𝐴
32a1i 9 . . 3 (⊤ → 𝑥𝐴)
4 nfralya.2 . . . 4 𝑥𝜑
54a1i 9 . . 3 (⊤ → Ⅎ𝑥𝜑)
61, 3, 5nfraldya 2525 . 2 (⊤ → Ⅎ𝑥𝑦𝐴 𝜑)
76mptru 1373 1 𝑥𝑦𝐴 𝜑
Colors of variables: wff set class
Syntax hints:  wtru 1365  wnf 1471  wnfc 2319  wral 2468
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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473
This theorem is referenced by:  nfiinya  3930  nfsup  7020  caucvgsrlemgt1  7823  axpre-suploclemres  7929  supinfneg  9624  infsupneg  9625  ctiunctlemudc  12487  trirec0  15246
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