Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dveel1 Structured version   Visualization version   GIF version

Theorem dveel1 2369
 Description: Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
Assertion
Ref Expression
dveel1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦𝑧 → ∀𝑥 𝑦𝑧))
Distinct variable group:   𝑥,𝑧

Proof of Theorem dveel1
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elequ1 1994 . 2 (𝑤 = 𝑦 → (𝑤𝑧𝑦𝑧))
21dvelimv 2337 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦𝑧 → ∀𝑥 𝑦𝑧))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1478 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-8 1989  ax-10 2016  ax-11 2031  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:  distel  31445
 Copyright terms: Public domain W3C validator