 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvdemo1 Structured version   Visualization version   GIF version

Theorem dvdemo1 4863
 Description: Demonstration of a theorem (scheme) that requires (meta)variables 𝑥 and 𝑦 to be distinct, but no others. It bundles the theorem schemes ∃𝑥(𝑥 = 𝑦 → 𝑥 ∈ 𝑥) and ∃𝑥(𝑥 = 𝑦 → 𝑦 ∈ 𝑥). Compare dvdemo2 4864. ("Bundles" is a term introduced by Raph Levien.) (Contributed by NM, 1-Dec-2006.)
Assertion
Ref Expression
dvdemo1 𝑥(𝑥 = 𝑦𝑧𝑥)
Distinct variable group:   𝑥,𝑦

Proof of Theorem dvdemo1
StepHypRef Expression
1 dtru 4817 . . 3 ¬ ∀𝑥 𝑥 = 𝑦
2 exnal 1751 . . 3 (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
31, 2mpbir 221 . 2 𝑥 ¬ 𝑥 = 𝑦
4 pm2.21 120 . 2 𝑥 = 𝑦 → (𝑥 = 𝑦𝑧𝑥))
53, 4eximii 1761 1 𝑥(𝑥 = 𝑦𝑧𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1478  ∃wex 1701 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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-nul 4749  ax-pow 4803 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