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Theorem nfeqf 2337
Description: A variable is effectively not free in an equality if it is not either of the involved variables. version of ax-c9 34494. (Contributed by Mario Carneiro, 6-Oct-2016.) Remove dependency on ax-11 2074. (Revised by Wolf Lammen, 6-Sep-2018.)
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 2069 . . 3 𝑧 ¬ ∀𝑧 𝑧 = 𝑥
2 nfna1 2069 . . 3 𝑧 ¬ ∀𝑧 𝑧 = 𝑦
31, 2nfan 1868 . 2 𝑧(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
4 equviniva 2004 . . 3 (𝑥 = 𝑦 → ∃𝑤(𝑥 = 𝑤𝑦 = 𝑤))
5 dveeq1 2336 . . . . . . . 8 (¬ ∀𝑧 𝑧 = 𝑥 → (𝑥 = 𝑤 → ∀𝑧 𝑥 = 𝑤))
65imp 444 . . . . . . 7 ((¬ ∀𝑧 𝑧 = 𝑥𝑥 = 𝑤) → ∀𝑧 𝑥 = 𝑤)
7 dveeq1 2336 . . . . . . . 8 (¬ ∀𝑧 𝑧 = 𝑦 → (𝑦 = 𝑤 → ∀𝑧 𝑦 = 𝑤))
87imp 444 . . . . . . 7 ((¬ ∀𝑧 𝑧 = 𝑦𝑦 = 𝑤) → ∀𝑧 𝑦 = 𝑤)
9 equtr2 2000 . . . . . . . 8 ((𝑥 = 𝑤𝑦 = 𝑤) → 𝑥 = 𝑦)
109alanimi 1784 . . . . . . 7 ((∀𝑧 𝑥 = 𝑤 ∧ ∀𝑧 𝑦 = 𝑤) → ∀𝑧 𝑥 = 𝑦)
116, 8, 10syl2an 493 . . . . . 6 (((¬ ∀𝑧 𝑧 = 𝑥𝑥 = 𝑤) ∧ (¬ ∀𝑧 𝑧 = 𝑦𝑦 = 𝑤)) → ∀𝑧 𝑥 = 𝑦)
1211an4s 886 . . . . 5 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) ∧ (𝑥 = 𝑤𝑦 = 𝑤)) → ∀𝑧 𝑥 = 𝑦)
1312ex 449 . . . 4 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → ((𝑥 = 𝑤𝑦 = 𝑤) → ∀𝑧 𝑥 = 𝑦))
1413exlimdv 1901 . . 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 2156 1 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wal 1521  wex 1744  wnf 1748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750
This theorem is referenced by:  axc9  2338  dvelimf  2365  equvel  2375  2ax6elem  2477  wl-exeq  33451  wl-nfeqfb  33453  wl-equsb4  33468  wl-2sb6d  33471  wl-sbalnae  33475
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