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Theorem nfeqf2 2333
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 1794 . 2 (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2 nfnf1 2071 . . 3 𝑥𝑥 𝑧 = 𝑦
3 ax13lem2 2332 . . . . 5 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦𝑧 = 𝑦))
4 ax13lem1 2284 . . . . 5 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
53, 4syld 47 . . . 4 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
6 df-nf 1750 . . . 4 (Ⅎ𝑥 𝑧 = 𝑦 ↔ (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
75, 6sylibr 224 . . 3 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
82, 7exlimi 2124 . 2 (∃𝑥 ¬ 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
91, 8sylbir 225 1 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1521  wex 1744  wnf 1748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750
This theorem is referenced by:  dveeq2  2334  nfeqf1  2335  sbal1  2488  copsexg  4985  axrepndlem1  9452  axpowndlem2  9458  axpowndlem3  9459  bj-dvelimdv  32959  bj-dvelimdv1  32960  wl-equsb3  33467  wl-sbcom2d-lem1  33472  wl-mo2df  33482  wl-eudf  33484  wl-euequ1f  33486
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