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Theorem nfrexd 2989
 Description: Deduction version of nfrex 2990. (Contributed by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
nfrexd.1 𝑦𝜑
nfrexd.2 (𝜑𝑥𝐴)
nfrexd.3 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfrexd (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)

Proof of Theorem nfrexd
StepHypRef Expression
1 dfrex2 2979 . 2 (∃𝑦𝐴 𝜓 ↔ ¬ ∀𝑦𝐴 ¬ 𝜓)
2 nfrexd.1 . . . 4 𝑦𝜑
3 nfrexd.2 . . . 4 (𝜑𝑥𝐴)
4 nfrexd.3 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
54nfnd 1769 . . . 4 (𝜑 → Ⅎ𝑥 ¬ 𝜓)
62, 3, 5nfrald 2928 . . 3 (𝜑 → Ⅎ𝑥𝑦𝐴 ¬ 𝜓)
76nfnd 1769 . 2 (𝜑 → Ⅎ𝑥 ¬ ∀𝑦𝐴 ¬ 𝜓)
81, 7nfxfrd 1772 1 (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  Ⅎwnf 1699  Ⅎwnfc 2738  ∀wral 2896  ∃wrex 2897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902 This theorem is referenced by:  nfrex  2990  nfunid  4379  nfiund  42219
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