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Theorem nfrexd 3373
Description: Deduction version of nfrex 3375. Usage of this theorem is discouraged because it depends on ax-13 2377. See nfrexdw 3310 for a version with a disjoint variable condition, but not requiring ax-13 2377. (Contributed by Mario Carneiro, 14-Oct-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfrald.1 𝑦𝜑
nfrald.2 (𝜑𝑥𝐴)
nfrald.3 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfrexd (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)

Proof of Theorem nfrexd
StepHypRef Expression
1 dfrex2 3073 . 2 (∃𝑦𝐴 𝜓 ↔ ¬ ∀𝑦𝐴 ¬ 𝜓)
2 nfrald.1 . . . 4 𝑦𝜑
3 nfrald.2 . . . 4 (𝜑𝑥𝐴)
4 nfrald.3 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
54nfnd 1858 . . . 4 (𝜑 → Ⅎ𝑥 ¬ 𝜓)
62, 3, 5nfrald 3372 . . 3 (𝜑 → Ⅎ𝑥𝑦𝐴 ¬ 𝜓)
76nfnd 1858 . 2 (𝜑 → Ⅎ𝑥 ¬ ∀𝑦𝐴 ¬ 𝜓)
81, 7nfxfrd 1854 1 (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wnf 1783  wnfc 2890  wral 3061  wrex 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-13 2377  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071
This theorem is referenced by:  nfrex  3375  nfiundg  49194
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