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Theorem nfrexd 3361
Description: Deduction version of nfrex 3363. Usage of this theorem is discouraged because it depends on ax-13 2363. See nfrexdw 3299 for a version with a disjoint variable condition, but not requiring ax-13 2363. (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 3065 . 2 (∃𝑦𝐴 𝜓 ↔ ¬ ∀𝑦𝐴 ¬ 𝜓)
2 nfrald.1 . . . 4 𝑦𝜑
3 nfrald.2 . . . 4 (𝜑𝑥𝐴)
4 nfrald.3 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
54nfnd 1853 . . . 4 (𝜑 → Ⅎ𝑥 ¬ 𝜓)
62, 3, 5nfrald 3360 . . 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 2875  wral 3053  wrex 3062
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-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-13 2363  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063
This theorem is referenced by:  nfrex  3363  nfiundg  47968
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