Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  eueq1 Structured version   Visualization version   GIF version

Theorem eueq1 3377
 Description: Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.)
Hypothesis
Ref Expression
eueq1.1 𝐴 ∈ V
Assertion
Ref Expression
eueq1 ∃!𝑥 𝑥 = 𝐴
Distinct variable group:   𝑥,𝐴

Proof of Theorem eueq1
StepHypRef Expression
1 eueq1.1 . 2 𝐴 ∈ V
2 eueq 3376 . 2 (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
31, 2mpbi 220 1 ∃!𝑥 𝑥 = 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1482   ∈ wcel 1989  ∃!weu 2469  Vcvv 3198 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3200 This theorem is referenced by:  eueq2  3378  eueq3  3379  fsn  6399  bj-nuliota  33003
 Copyright terms: Public domain W3C validator