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Theorem nfrexdg 3267
 Description: Deduction version of nfrexg 3269. Usage of this theorem is discouraged because it depends on ax-13 2379. See nfrexd 3266 for a version with a disjoint variable condition, but not requiring ax-13 2379. (Contributed by Mario Carneiro, 14-Oct-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfrexdg.1 𝑦𝜑
nfrexdg.2 (𝜑𝑥𝐴)
nfrexdg.3 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfrexdg (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)

Proof of Theorem nfrexdg
StepHypRef Expression
1 dfrex2 3202 . 2 (∃𝑦𝐴 𝜓 ↔ ¬ ∀𝑦𝐴 ¬ 𝜓)
2 nfrexdg.1 . . . 4 𝑦𝜑
3 nfrexdg.2 . . . 4 (𝜑𝑥𝐴)
4 nfrexdg.3 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
54nfnd 1859 . . . 4 (𝜑 → Ⅎ𝑥 ¬ 𝜓)
62, 3, 5nfrald 3188 . . 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-13 2379  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  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:  nfrexg  3269  nfiundg  45246
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