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Theorem nfeqf 2300
Description: A variable is effectively not free in an equality if it is not either of the involved variables. version of ax-c9 33652. (Contributed by Mario Carneiro, 6-Oct-2016.) Remove dependency on ax-11 2031. (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 2026 . . 3 𝑧 ¬ ∀𝑧 𝑧 = 𝑥
2 nfna1 2026 . . 3 𝑧 ¬ ∀𝑧 𝑧 = 𝑦
31, 2nfan 1825 . 2 𝑧(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
4 equviniva 1957 . . 3 (𝑥 = 𝑦 → ∃𝑤(𝑥 = 𝑤𝑦 = 𝑤))
5 dveeq1 2299 . . . . . . . 8 (¬ ∀𝑧 𝑧 = 𝑥 → (𝑥 = 𝑤 → ∀𝑧 𝑥 = 𝑤))
65imp 445 . . . . . . 7 ((¬ ∀𝑧 𝑧 = 𝑥𝑥 = 𝑤) → ∀𝑧 𝑥 = 𝑤)
7 dveeq1 2299 . . . . . . . 8 (¬ ∀𝑧 𝑧 = 𝑦 → (𝑦 = 𝑤 → ∀𝑧 𝑦 = 𝑤))
87imp 445 . . . . . . 7 ((¬ ∀𝑧 𝑧 = 𝑦𝑦 = 𝑤) → ∀𝑧 𝑦 = 𝑤)
9 equtr2 1951 . . . . . . . 8 ((𝑥 = 𝑤𝑦 = 𝑤) → 𝑥 = 𝑦)
109alanimi 1741 . . . . . . 7 ((∀𝑧 𝑥 = 𝑤 ∧ ∀𝑧 𝑦 = 𝑤) → ∀𝑧 𝑥 = 𝑦)
116, 8, 10syl2an 494 . . . . . 6 (((¬ ∀𝑧 𝑧 = 𝑥𝑥 = 𝑤) ∧ (¬ ∀𝑧 𝑧 = 𝑦𝑦 = 𝑤)) → ∀𝑧 𝑥 = 𝑦)
1211an4s 868 . . . . 5 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) ∧ (𝑥 = 𝑤𝑦 = 𝑤)) → ∀𝑧 𝑥 = 𝑦)
1312ex 450 . . . 4 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → ((𝑥 = 𝑤𝑦 = 𝑤) → ∀𝑧 𝑥 = 𝑦))
1413exlimdv 1858 . . 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 2115 1 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  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-tru 1483  df-ex 1702  df-nf 1707
This theorem is referenced by:  axc9  2301  dvelimf  2333  equvel  2346  2ax6elem  2448  wl-exeq  32950  wl-nfeqfb  32952  wl-equsb4  32967  wl-2sb6d  32970  wl-sbalnae  32974
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