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Theorem nfeqf2 2372
Description: An equation between setvar is free of any other setvar. Usage of this theorem is discouraged because it depends on ax-13 2367. (Contributed by Wolf Lammen, 9-Jun-2019.) Remove dependency on ax-12 2167. (Revised by Wolf Lammen, 16-Dec-2022.) (New usage is discouraged.)
Assertion
Ref Expression
nfeqf2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
Distinct variable group:   𝑥,𝑧

Proof of Theorem nfeqf2
StepHypRef Expression
1 exnal 1822 . 2 (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2 hbe1 2132 . . . . 5 (∃𝑥 𝑧 = 𝑦 → ∀𝑥𝑥 𝑧 = 𝑦)
3 ax13lem2 2371 . . . . . 6 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦𝑧 = 𝑦))
4 ax13lem1 2369 . . . . . 6 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
53, 4syldc 48 . . . . 5 (∃𝑥 𝑧 = 𝑦 → (¬ 𝑥 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
62, 5eximdh 1860 . . . 4 (∃𝑥 𝑧 = 𝑦 → (∃𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥𝑥 𝑧 = 𝑦))
7 hbe1a 2133 . . . 4 (∃𝑥𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)
86, 7syl6com 37 . . 3 (∃𝑥 ¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
98nfd 1785 . 2 (∃𝑥 ¬ 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
101, 9sylbir 234 1 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1532  wex 1774  wnf 1778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-10 2130  ax-13 2367
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-nf 1779
This theorem is referenced by:  dveeq2  2373  nfeqf1  2374  sb4b  2470  sbal1  2523  copsexg  5493  axrepndlem1  10616  axpowndlem2  10622  axpowndlem3  10623  bj-dvelimdv  36328  bj-dvelimdv1  36329  wl-equsb3  37023  wl-sbcom2d-lem1  37026  wl-mo2df  37037  wl-eudf  37039  wl-euequf  37041
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