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Theorem dvdemo1 4824
 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 4825. ("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 4778 . . 3 ¬ ∀𝑥 𝑥 = 𝑦
2 exnal 1743 . . 3 (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
31, 2mpbir 219 . 2 𝑥 ¬ 𝑥 = 𝑦
4 pm2.21 118 . 2 𝑥 = 𝑦 → (𝑥 = 𝑦𝑧𝑥))
53, 4eximii 1753 1 𝑥(𝑥 = 𝑦𝑧𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1472  ∃wex 1694 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-nul 4712  ax-pow 4764 This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700 This theorem is referenced by: (None)
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