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

Theorem dveel1 2484
Description: Quantifier introduction when one pair of variables is distinct. Usage of this theorem is discouraged because it depends on ax-13 2390. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)
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 2121 . 2 (𝑤 = 𝑦 → (𝑤𝑧𝑦𝑧))
21dvelimv 2474 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦𝑧 → ∀𝑥 𝑦𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785
This theorem is referenced by:  distel  33056
  Copyright terms: Public domain W3C validator