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Theorem nfrexd 3345
Description: Deduction version of nfrex 3347. Usage of this theorem is discouraged because it depends on ax-13 2371. See nfrexdw 3292 for a version with a disjoint variable condition, but not requiring ax-13 2371. (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 1862 . . . 4 (𝜑 → Ⅎ𝑥 ¬ 𝜓)
62, 3, 5nfrald 3344 . . 3 (𝜑 → Ⅎ𝑥𝑦𝐴 ¬ 𝜓)
76nfnd 1862 . 2 (𝜑 → Ⅎ𝑥 ¬ ∀𝑦𝐴 ¬ 𝜓)
81, 7nfxfrd 1857 1 (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wnf 1786  wnfc 2884  wral 3061  wrex 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-13 2371  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071
This theorem is referenced by:  nfrex  3347  nfiundg  47206
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