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Theorem nfraldw 2489
Description: Not-free for restricted universal quantification where 𝑥 and 𝑦 are distinct. See nfraldya 2492 for a version with 𝑦 and 𝐴 distinct instead. (Contributed by NM, 15-Feb-2013.) (Revised by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
nfraldw.1 𝑦𝜑
nfraldw.2 (𝜑𝑥𝐴)
nfraldw.3 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfraldw (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfraldw
StepHypRef Expression
1 df-ral 2440 . 2 (∀𝑦𝐴 𝜓 ↔ ∀𝑦(𝑦𝐴𝜓))
2 nfraldw.1 . . 3 𝑦𝜑
3 nfcvd 2300 . . . . 5 (𝜑𝑥𝑦)
4 nfraldw.2 . . . . 5 (𝜑𝑥𝐴)
53, 4nfeld 2315 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐴)
6 nfraldw.3 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
75, 6nfimd 1565 . . 3 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
82, 7nfald 1740 . 2 (𝜑 → Ⅎ𝑥𝑦(𝑦𝐴𝜓))
91, 8nfxfrd 1455 1 (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1333  wnf 1440  wcel 2128  wnfc 2286  wral 2435
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-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440
This theorem is referenced by:  nfralw  2494
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