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Theorem nfeqf2 2339
Description: An equation between setvar is free of any other setvar. (Contributed by Wolf Lammen, 9-Jun-2019.) Remove dependency on ax-12 2163. (Revised by Wolf Lammen, 16-Dec-2022.)
Assertion
Ref Expression
nfeqf2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
Distinct variable group:   𝑥,𝑧

Proof of Theorem nfeqf2
StepHypRef Expression
1 exnal 1870 . 2 (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2 hbe1 2137 . . . . 5 (∃𝑥 𝑧 = 𝑦 → ∀𝑥𝑥 𝑧 = 𝑦)
3 ax13lem2 2338 . . . . . 6 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦𝑧 = 𝑦))
4 ax13lem1 2336 . . . . . 6 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
53, 4syldc 48 . . . . 5 (∃𝑥 𝑧 = 𝑦 → (¬ 𝑥 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
62, 5eximdh 1909 . . . 4 (∃𝑥 𝑧 = 𝑦 → (∃𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥𝑥 𝑧 = 𝑦))
7 hbe1a 2138 . . . 4 (∃𝑥𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)
86, 7syl6com 37 . . 3 (∃𝑥 ¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
98nfd 1834 . 2 (∃𝑥 ¬ 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
101, 9sylbir 227 1 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1599  wex 1823  wnf 1827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-13 2334
This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-nf 1828
This theorem is referenced by:  dveeq2  2342  nfeqf1  2343  sbal1  2539  copsexg  5189  axrepndlem1  9751  axpowndlem2  9757  axpowndlem3  9758  bj-dvelimdv  33412  bj-dvelimdv1  33413  wl-equsb3  33935  wl-sbcom2d-lem1  33939  wl-mo2df  33949  wl-eudf  33951  wl-euequf  33953
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