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Theorem dvdemo1 5085
 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 5086. ("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 5082 . . 3 ¬ ∀𝑥 𝑥 = 𝑦
2 exnal 1870 . . 3 (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
31, 2mpbir 223 . 2 𝑥 ¬ 𝑥 = 𝑦
4 pm2.21 121 . 2 𝑥 = 𝑦 → (𝑥 = 𝑦𝑧𝑥))
53, 4eximii 1880 1 𝑥(𝑥 = 𝑦𝑧𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1599  ∃wex 1823 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-12 2162  ax-13 2333  ax-nul 5025  ax-pow 5077 This theorem depends on definitions:  df-bi 199  df-an 387  df-tru 1605  df-ex 1824  df-nf 1828 This theorem is referenced by: (None)
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