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Theorem nfraldxy 2523
Description: Old name for nfraldw 2522. (Contributed by Jim Kingdon, 29-May-2018.) (New usage is discouraged.)
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 2473 . 2 (∀𝑦𝐴 𝜓 ↔ ∀𝑦(𝑦𝐴𝜓))
2 nfraldxy.2 . . 3 𝑦𝜑
3 nfcv 2332 . . . . . 6 𝑥𝑦
43a1i 9 . . . . 5 (𝜑𝑥𝑦)
5 nfraldxy.3 . . . . 5 (𝜑𝑥𝐴)
64, 5nfeld 2348 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐴)
7 nfraldxy.4 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
86, 7nfimd 1596 . . 3 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
92, 8nfald 1771 . 2 (𝜑 → Ⅎ𝑥𝑦(𝑦𝐴𝜓))
101, 9nfxfrd 1486 1 (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1362  wnf 1471  wcel 2160  wnfc 2319  wral 2468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473
This theorem is referenced by:  nfraldya  2525  nfralxy  2528  strcollnft  15194
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