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Theorem nfralya 2359
Description: Not-free for restricted universal quantification where 𝑦 and 𝐴 are distinct. See nfralxy 2357 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 1355 . . 3 𝑦
2 nfralya.1 . . . 4 𝑥𝐴
32a1i 9 . . 3 (⊤ → 𝑥𝐴)
4 nfralya.2 . . . 4 𝑥𝜑
54a1i 9 . . 3 (⊤ → Ⅎ𝑥𝜑)
61, 3, 5nfraldya 2355 . 2 (⊤ → Ⅎ𝑥𝑦𝐴 𝜑)
76trud 1252 1 𝑥𝑦𝐴 𝜑
Colors of variables: wff set class
Syntax hints:  wtru 1244  wnf 1349  wnfc 2165  wral 2303
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308
This theorem is referenced by:  nfiinya  3683  caucvgsrlemgt1  6836
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