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Theorem eueq 2784
Description: Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)
Assertion
Ref Expression
eueq (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem eueq
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqtr3 2107 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐴) → 𝑥 = 𝑦)
21gen2 1384 . . 3 𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐴) → 𝑥 = 𝑦)
32biantru 296 . 2 (∃𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐴) → 𝑥 = 𝑦)))
4 isset 2625 . 2 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
5 eqeq1 2094 . . 3 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
65eu4 2010 . 2 (∃!𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐴) → 𝑥 = 𝑦)))
73, 4, 63bitr4i 210 1 (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1287   = wceq 1289  wex 1426  wcel 1438  ∃!weu 1948  Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621
This theorem is referenced by:  eueq1  2785  moeq  2788  mosubt  2790  reuhypd  4284  mptfng  5125
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