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Theorem eueqOLD 3589
 Description: Obsolete proof of eueq 3588 as of 24-Sep-2022. (Contributed by NM, 25-Nov-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
eueqOLD (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem eueqOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqtr3 2800 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐴) → 𝑥 = 𝑦)
21gen2 1840 . . 3 𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐴) → 𝑥 = 𝑦)
32biantru 525 . 2 (∃𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐴) → 𝑥 = 𝑦)))
4 isset 3408 . 2 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
5 eqeq1 2781 . . 3 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
65eu4 2649 . 2 (∃!𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐴) → 𝑥 = 𝑦)))
73, 4, 63bitr4i 295 1 (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1599   = wceq 1601  ∃wex 1823   ∈ wcel 2106  ∃!weu 2585  Vcvv 3397 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-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-v 3399 This theorem is referenced by:  moeqOLD  3593
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