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Theorem eueqi 3698
Description: There exists a unique set equal to a given set. Inference associated with euequ 2583. See euequ 2583 in the case of a setvar. (Contributed by NM, 5-Apr-1995.)
Hypothesis
Ref Expression
eueqi.1 𝐴 ∈ V
Assertion
Ref Expression
eueqi ∃!𝑥 𝑥 = 𝐴
Distinct variable group:   𝑥,𝐴

Proof of Theorem eueqi
StepHypRef Expression
1 eueqi.1 . 2 𝐴 ∈ V
2 eueq 3697 . 2 (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
31, 2mpbi 229 1 ∃!𝑥 𝑥 = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  ∃!weu 2554  Vcvv 3466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468
This theorem is referenced by:  eueq2  3699  eueq3  3700  fsn  7126  bj-nuliota  36439  prprval  46728
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