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Theorem nfeqf2 2285
 Description: An equation between setvar is free of any other setvar. (Contributed by Wolf Lammen, 9-Jun-2019.)
Assertion
Ref Expression
nfeqf2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
Distinct variable group:   𝑥,𝑧

Proof of Theorem nfeqf2
StepHypRef Expression
1 exnal 1744 . 2 (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2 nfnf1 2018 . . 3 𝑥𝑥 𝑧 = 𝑦
3 ax13lem2 2284 . . . . 5 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦𝑧 = 𝑦))
4 ax13lem1 2236 . . . . 5 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
53, 4syld 46 . . . 4 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
6 df-nf 1701 . . . 4 (Ⅎ𝑥 𝑧 = 𝑦 ↔ (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
75, 6sylibr 223 . . 3 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
82, 7exlimi 2073 . 2 (∃𝑥 ¬ 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
91, 8sylbir 224 1 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701 This theorem is referenced by:  dveeq2  2286  nfeqf1  2287  sbal1  2448  copsexg  4882  axrepndlem1  9293  axpowndlem2  9299  axpowndlem3  9300  bj-dvelimdv  32027  bj-dvelimdv1  32028  wl-equsb3  32516  wl-sbcom2d-lem1  32521  wl-mo2df  32531  wl-eudf  32533  wl-euequ1f  32535
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