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Theorem nfeqf2 2296
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 1751 . 2 (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2 nfnf1 2028 . . 3 𝑥𝑥 𝑧 = 𝑦
3 ax13lem2 2295 . . . . 5 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦𝑧 = 𝑦))
4 ax13lem1 2247 . . . . 5 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
53, 4syld 47 . . . 4 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
6 df-nf 1707 . . . 4 (Ⅎ𝑥 𝑧 = 𝑦 ↔ (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
75, 6sylibr 224 . . 3 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
82, 7exlimi 2084 . 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 1478  wex 1701  wnf 1705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707
This theorem is referenced by:  dveeq2  2297  nfeqf1  2298  sbal1  2459  copsexg  4916  axrepndlem1  9358  axpowndlem2  9364  axpowndlem3  9365  bj-dvelimdv  32476  bj-dvelimdv1  32477  wl-equsb3  32966  wl-sbcom2d-lem1  32971  wl-mo2df  32981  wl-eudf  32983  wl-euequ1f  32985
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