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Theorem nfrexd 3266
 Description: Deduction version of nfrex 3268. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2379. See nfrexdg 3267 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
Hypotheses
Ref Expression
nfrexd.1 𝑦𝜑
nfrexd.2 (𝜑𝑥𝐴)
nfrexd.3 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfrexd (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfrexd
StepHypRef Expression
1 dfrex2 3202 . 2 (∃𝑦𝐴 𝜓 ↔ ¬ ∀𝑦𝐴 ¬ 𝜓)
2 nfrexd.1 . . . 4 𝑦𝜑
3 nfrexd.2 . . . 4 (𝜑𝑥𝐴)
4 nfrexd.3 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
54nfnd 1859 . . . 4 (𝜑 → Ⅎ𝑥 ¬ 𝜓)
62, 3, 5nfraldw 3187 . . 3 (𝜑 → Ⅎ𝑥𝑦𝐴 ¬ 𝜓)
76nfnd 1859 . 2 (𝜑 → Ⅎ𝑥 ¬ ∀𝑦𝐴 ¬ 𝜓)
81, 7nfxfrd 1855 1 (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  Ⅎwnf 1785  Ⅎwnfc 2936  ∀wral 3106  ∃wrex 3107 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112 This theorem is referenced by:  nfrex  3268  nfunid  4807  nfiund  45245
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