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Theorem nfeqf 2388
 Description: A variable is effectively not free in an equality if it is not either of the involved variables. Ⅎ version of ax-c9 36337. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by Mario Carneiro, 6-Oct-2016.) Remove dependency on ax-11 2158. (Revised by Wolf Lammen, 6-Sep-2018.) (New usage is discouraged.)
Assertion
Ref Expression
nfeqf ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦)

Proof of Theorem nfeqf
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfna1 2153 . . 3 𝑧 ¬ ∀𝑧 𝑧 = 𝑥
2 nfna1 2153 . . 3 𝑧 ¬ ∀𝑧 𝑧 = 𝑦
31, 2nfan 1900 . 2 𝑧(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
4 equvinva 2037 . . 3 (𝑥 = 𝑦 → ∃𝑤(𝑥 = 𝑤𝑦 = 𝑤))
5 dveeq1 2387 . . . . . . . 8 (¬ ∀𝑧 𝑧 = 𝑥 → (𝑥 = 𝑤 → ∀𝑧 𝑥 = 𝑤))
65imp 410 . . . . . . 7 ((¬ ∀𝑧 𝑧 = 𝑥𝑥 = 𝑤) → ∀𝑧 𝑥 = 𝑤)
7 dveeq1 2387 . . . . . . . 8 (¬ ∀𝑧 𝑧 = 𝑦 → (𝑦 = 𝑤 → ∀𝑧 𝑦 = 𝑤))
87imp 410 . . . . . . 7 ((¬ ∀𝑧 𝑧 = 𝑦𝑦 = 𝑤) → ∀𝑧 𝑦 = 𝑤)
9 equtr2 2034 . . . . . . . 8 ((𝑥 = 𝑤𝑦 = 𝑤) → 𝑥 = 𝑦)
109alanimi 1818 . . . . . . 7 ((∀𝑧 𝑥 = 𝑤 ∧ ∀𝑧 𝑦 = 𝑤) → ∀𝑧 𝑥 = 𝑦)
116, 8, 10syl2an 598 . . . . . 6 (((¬ ∀𝑧 𝑧 = 𝑥𝑥 = 𝑤) ∧ (¬ ∀𝑧 𝑧 = 𝑦𝑦 = 𝑤)) → ∀𝑧 𝑥 = 𝑦)
1211an4s 659 . . . . 5 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) ∧ (𝑥 = 𝑤𝑦 = 𝑤)) → ∀𝑧 𝑥 = 𝑦)
1312ex 416 . . . 4 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → ((𝑥 = 𝑤𝑦 = 𝑤) → ∀𝑧 𝑥 = 𝑦))
1413exlimdv 1934 . . 3 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∃𝑤(𝑥 = 𝑤𝑦 = 𝑤) → ∀𝑧 𝑥 = 𝑦))
154, 14syl5 34 . 2 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
163, 15nf5d 2288 1 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1536  ∃wex 1781  Ⅎwnf 1785 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175  ax-13 2379 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786 This theorem is referenced by:  axc9  2389  dvelimf  2459  equvel  2468  2ax6elem  2482  wl-exeq  35090  wl-nfeqfb  35092  wl-equsb4  35109  wl-2sb6d  35110  wl-sbalnae  35114
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