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Theorem nfrexd 3153
Description: Deduction version of nfrex 3154. (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 3143 . 2 (∃𝑦𝐴 𝜓 ↔ ¬ ∀𝑦𝐴 ¬ 𝜓)
2 nfrexd.1 . . . 4 𝑦𝜑
3 nfrexd.2 . . . 4 (𝜑𝑥𝐴)
4 nfrexd.3 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
54nfnd 1935 . . . 4 (𝜑 → Ⅎ𝑥 ¬ 𝜓)
62, 3, 5nfrald 3092 . . 3 (𝜑 → Ⅎ𝑥𝑦𝐴 ¬ 𝜓)
76nfnd 1935 . 2 (𝜑 → Ⅎ𝑥 ¬ ∀𝑦𝐴 ¬ 𝜓)
81, 7nfxfrd 1929 1 (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wnf 1855  wnfc 2899  wral 3060  wrex 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066
This theorem is referenced by:  nfrex  3154  nfunid  4579  nfiund  42939
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