MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dveeq2 Structured version   Visualization version   GIF version

Theorem dveeq2 2377
Description: Quantifier introduction when one pair of variables is distinct. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.) Remove dependency on ax-11 2158. (Revised by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.)
Assertion
Ref Expression
dveeq2 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Distinct variable group:   𝑥,𝑧

Proof of Theorem dveeq2
StepHypRef Expression
1 nfeqf2 2376 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
21nf5rd 2194 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-10 2141  ax-12 2175  ax-13 2371
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-nf 1792
This theorem is referenced by:  axc15  2421
  Copyright terms: Public domain W3C validator