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Theorem nfralya 2476
Description: Not-free for restricted universal quantification where 𝑦 and 𝐴 are distinct. See nfralxy 2474 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 1443 . . 3 𝑦
2 nfralya.1 . . . 4 𝑥𝐴
32a1i 9 . . 3 (⊤ → 𝑥𝐴)
4 nfralya.2 . . . 4 𝑥𝜑
54a1i 9 . . 3 (⊤ → Ⅎ𝑥𝜑)
61, 3, 5nfraldya 2472 . 2 (⊤ → Ⅎ𝑥𝑦𝐴 𝜑)
76mptru 1341 1 𝑥𝑦𝐴 𝜑
Colors of variables: wff set class
Syntax hints:  wtru 1333  wnf 1437  wnfc 2269  wral 2417
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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422
This theorem is referenced by:  nfiinya  3849  nfsup  6886  caucvgsrlemgt1  7626  axpre-suploclemres  7732  supinfneg  9416  infsupneg  9417  ctiunctlemudc  11984  trirec0  13410
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