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Theorem nfralya 2410
 Description: Not-free for restricted universal quantification where 𝑦 and 𝐴 are distinct. See nfralxy 2408 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 1396 . . 3 𝑦
2 nfralya.1 . . . 4 𝑥𝐴
32a1i 9 . . 3 (⊤ → 𝑥𝐴)
4 nfralya.2 . . . 4 𝑥𝜑
54a1i 9 . . 3 (⊤ → Ⅎ𝑥𝜑)
61, 3, 5nfraldya 2406 . 2 (⊤ → Ⅎ𝑥𝑦𝐴 𝜑)
76trud 1294 1 𝑥𝑦𝐴 𝜑
 Colors of variables: wff set class Syntax hints:  ⊤wtru 1286  Ⅎwnf 1390  Ⅎwnfc 2210  ∀wral 2353 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358 This theorem is referenced by:  nfiinya  3728  nfsup  6507  caucvgsrlemgt1  7119  supinfneg  8850  infsupneg  8851
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