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Theorem nfraldxy 2421
Description: Not-free for restricted universal quantification where 𝑥 and 𝑦 are distinct. See nfraldya 2423 for a version with 𝑦 and 𝐴 distinct instead. (Contributed by Jim Kingdon, 29-May-2018.)
Hypotheses
Ref Expression
nfraldxy.2 𝑦𝜑
nfraldxy.3 (𝜑𝑥𝐴)
nfraldxy.4 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfraldxy (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfraldxy
StepHypRef Expression
1 df-ral 2375 . 2 (∀𝑦𝐴 𝜓 ↔ ∀𝑦(𝑦𝐴𝜓))
2 nfraldxy.2 . . 3 𝑦𝜑
3 nfcv 2235 . . . . . 6 𝑥𝑦
43a1i 9 . . . . 5 (𝜑𝑥𝑦)
5 nfraldxy.3 . . . . 5 (𝜑𝑥𝐴)
64, 5nfeld 2251 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐴)
7 nfraldxy.4 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
86, 7nfimd 1529 . . 3 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
92, 8nfald 1697 . 2 (𝜑 → Ⅎ𝑥𝑦(𝑦𝐴𝜓))
101, 9nfxfrd 1416 1 (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1294  wnf 1401  wcel 1445  wnfc 2222  wral 2370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-4 1452  ax-17 1471  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-nf 1402  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375
This theorem is referenced by:  nfraldya  2423  nfralxy  2425
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