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Theorem eueq 3530
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 2792 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐴) → 𝑥 = 𝑦)
21gen2 1871 . . 3 𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐴) → 𝑥 = 𝑦)
32biantru 519 . 2 (∃𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐴) → 𝑥 = 𝑦)))
4 isset 3359 . 2 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
5 eqeq1 2775 . . 3 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
65eu4 2667 . 2 (∃!𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐴) → 𝑥 = 𝑦)))
73, 4, 63bitr4i 292 1 (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1629   = wceq 1631  wex 1852  wcel 2145  ∃!weu 2618  Vcvv 3351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-v 3353
This theorem is referenced by:  eueq1  3531  moeq  3534  reuhypd  5023  mptfnf  6155  mptfng  6159  upxp  21647  iotasbc  39146  sprval  42257
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