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Theorem nfeqf2 2464
Description: An equation between setvar is free of any other setvar. (Contributed by Wolf Lammen, 9-Jun-2019.) Remove dependency on ax-12 2214. (Revised by Wolf Lammen, 20-Jul-2022.)
Assertion
Ref Expression
nfeqf2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
Distinct variable group:   𝑥,𝑧

Proof of Theorem nfeqf2
StepHypRef Expression
1 exnal 1911 . 2 (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2 hbe1 2187 . . . 4 (∃𝑥 𝑧 = 𝑦 → ∀𝑥𝑥 𝑧 = 𝑦)
3 ax13lem2 2463 . . . . . . 7 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦𝑧 = 𝑦))
4 ax13lem1 2422 . . . . . . 7 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
53, 4syldc 48 . . . . . 6 (∃𝑥 𝑧 = 𝑦 → (¬ 𝑥 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
65aleximi 1916 . . . . 5 (∀𝑥𝑥 𝑧 = 𝑦 → (∃𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥𝑥 𝑧 = 𝑦))
76com12 32 . . . 4 (∃𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝑥 𝑧 = 𝑦 → ∃𝑥𝑥 𝑧 = 𝑦))
8 hbe1a 2188 . . . 4 (∃𝑥𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)
92, 7, 8syl56 36 . . 3 (∃𝑥 ¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
109nfd 1870 . 2 (∃𝑥 ¬ 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
111, 10sylbir 226 1 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1635  wex 1859  wnf 1863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-10 2185  ax-13 2420
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1860  df-nf 1864
This theorem is referenced by:  dveeq2  2466  nfeqf1  2467  sbal1  2620  copsexg  5145  axrepndlem1  9695  axpowndlem2  9701  axpowndlem3  9702  bj-dvelimdv  33144  bj-dvelimdv1  33145  wl-equsb3  33650  wl-sbcom2d-lem1  33654  wl-mo2df  33664  wl-eudf  33666  wl-euequif  33668
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