MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfeqf Structured version   Visualization version   GIF version

Theorem nfeqf 2388
Description: A variable is effectively not free in an equality if it is not either of the involved variables. version of ax-c9 34904. (Contributed by Mario Carneiro, 6-Oct-2016.) Remove dependency on ax-11 2200. (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 2196 . . 3 𝑧 ¬ ∀𝑧 𝑧 = 𝑥
2 nfna1 2196 . . 3 𝑧 ¬ ∀𝑧 𝑧 = 𝑦
31, 2nfan 1999 . 2 𝑧(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
4 equvinva 2131 . . 3 (𝑥 = 𝑦 → ∃𝑤(𝑥 = 𝑤𝑦 = 𝑤))
5 dveeq1 2387 . . . . . . . 8 (¬ ∀𝑧 𝑧 = 𝑥 → (𝑥 = 𝑤 → ∀𝑧 𝑥 = 𝑤))
65imp 396 . . . . . . 7 ((¬ ∀𝑧 𝑧 = 𝑥𝑥 = 𝑤) → ∀𝑧 𝑥 = 𝑤)
7 dveeq1 2387 . . . . . . . 8 (¬ ∀𝑧 𝑧 = 𝑦 → (𝑦 = 𝑤 → ∀𝑧 𝑦 = 𝑤))
87imp 396 . . . . . . 7 ((¬ ∀𝑧 𝑧 = 𝑦𝑦 = 𝑤) → ∀𝑧 𝑦 = 𝑤)
9 equtr2 2126 . . . . . . . 8 ((𝑥 = 𝑤𝑦 = 𝑤) → 𝑥 = 𝑦)
109alanimi 1912 . . . . . . 7 ((∀𝑧 𝑥 = 𝑤 ∧ ∀𝑧 𝑦 = 𝑤) → ∀𝑧 𝑥 = 𝑦)
116, 8, 10syl2an 590 . . . . . 6 (((¬ ∀𝑧 𝑧 = 𝑥𝑥 = 𝑤) ∧ (¬ ∀𝑧 𝑧 = 𝑦𝑦 = 𝑤)) → ∀𝑧 𝑥 = 𝑦)
1211an4s 651 . . . . 5 (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) ∧ (𝑥 = 𝑤𝑦 = 𝑤)) → ∀𝑧 𝑥 = 𝑦)
1312ex 402 . . . 4 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → ((𝑥 = 𝑤𝑦 = 𝑤) → ∀𝑧 𝑥 = 𝑦))
1413exlimdv 2029 . . 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 2307 1 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385  wal 1651  wex 1875  wnf 1879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-10 2185  ax-12 2213  ax-13 2377
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880
This theorem is referenced by:  axc9  2389  dvelimf  2455  equvel  2463  2ax6elem  2569  wl-exeq  33804  wl-nfeqfb  33806  wl-equsb4  33822  wl-2sb6d  33824  wl-sbalnae  33828
  Copyright terms: Public domain W3C validator