Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dveel2 Structured version   Visualization version   GIF version

Theorem dveel2 2518
 Description: Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
Assertion
Ref Expression
dveel2 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧𝑦 → ∀𝑥 𝑧𝑦))
Distinct variable group:   𝑥,𝑧

Proof of Theorem dveel2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elequ2 2159 . 2 (𝑤 = 𝑦 → (𝑧𝑤𝑧𝑦))
21dvelimv 2488 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧𝑦 → ∀𝑥 𝑧𝑦))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1629 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858 This theorem is referenced by:  axc14  2519
 Copyright terms: Public domain W3C validator