Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dveeq1-o Structured version   Visualization version   GIF version

Theorem dveeq1-o 34641
Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq1 2409 using ax-c11 . (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dveeq1-o (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Distinct variable group:   𝑥,𝑧

Proof of Theorem dveeq1-o
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ax-5 1952 . 2 (𝑤 = 𝑧 → ∀𝑥 𝑤 = 𝑧)
2 ax-5 1952 . 2 (𝑦 = 𝑧 → ∀𝑤 𝑦 = 𝑧)
3 equequ1 2071 . 2 (𝑤 = 𝑦 → (𝑤 = 𝑧𝑦 = 𝑧))
41, 2, 3dvelimf-o 34635 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-c5 34589  ax-c4 34590  ax-c7 34591  ax-c10 34592  ax-c11 34593  ax-c9 34596
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823
This theorem is referenced by:  ax12inda2ALT  34652
  Copyright terms: Public domain W3C validator