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Theorem nfrexdw 3301
Description: Deduction version of nfrexw 3304. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2365. See nfrexd 3363 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
Hypotheses
Ref Expression
nfraldw.1 𝑦𝜑
nfraldw.2 (𝜑𝑥𝐴)
nfraldw.3 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfrexdw (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfrexdw
StepHypRef Expression
1 dfrex2 3067 . 2 (∃𝑦𝐴 𝜓 ↔ ¬ ∀𝑦𝐴 ¬ 𝜓)
2 nfraldw.1 . . . 4 𝑦𝜑
3 nfraldw.2 . . . 4 (𝜑𝑥𝐴)
4 nfraldw.3 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
54nfnd 1853 . . . 4 (𝜑 → Ⅎ𝑥 ¬ 𝜓)
62, 3, 5nfraldw 3300 . . 3 (𝜑 → Ⅎ𝑥𝑦𝐴 ¬ 𝜓)
76nfnd 1853 . 2 (𝜑 → Ⅎ𝑥 ¬ ∀𝑦𝐴 ¬ 𝜓)
81, 7nfxfrd 1848 1 (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wnf 1777  wnfc 2877  wral 3055  wrex 3064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-10 2129  ax-11 2146  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-nf 1778  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065
This theorem is referenced by:  nfrexw  3304  nfunid  4908  nfttrcld  9704  nfiund  47974
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