Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dveeq2ALT Structured version   Visualization version   GIF version

Theorem dveeq2ALT 2339
 Description: Alternate proof of dveeq2 2297, shorter but requiring ax-11 2031. (Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
dveeq2ALT (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Distinct variable group:   𝑥,𝑧

Proof of Theorem dveeq2ALT
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 equequ2 1950 . 2 (𝑤 = 𝑦 → (𝑧 = 𝑤𝑧 = 𝑦))
21dvelimv 2337 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
 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-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: (None)
 Copyright terms: Public domain W3C validator