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Theorem nfraldw 3223
Description: Deduction version of nfralw 3225. Version of nfrald 3224 with a disjoint variable condition, which does not require ax-13 2386. (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 3143 . 2 (∀𝑦𝐴 𝜓 ↔ ∀𝑦(𝑦𝐴𝜓))
2 nfraldw.1 . . 3 𝑦𝜑
3 nfcvd 2978 . . . . 5 (𝜑𝑥𝑦)
4 nfraldw.2 . . . . 5 (𝜑𝑥𝐴)
53, 4nfeld 2989 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐴)
6 nfraldw.3 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
75, 6nfimd 1891 . . 3 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
82, 7nfald 2343 . 2 (𝜑 → Ⅎ𝑥𝑦(𝑦𝐴𝜓))
91, 8nfxfrd 1850 1 (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531  wnf 1780  wcel 2110  wnfc 2961  wral 3138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143
This theorem is referenced by:  nfralw  3225  nfrexd  3307
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