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Theorem eueqi 3718
Description: There exists a unique set equal to a given set. Inference associated with euequ 2595. See euequ 2595 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 3717 . 2 (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
31, 2mpbi 230 1 ∃!𝑥 𝑥 = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  ∃!weu 2566  Vcvv 3478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480
This theorem is referenced by:  eueq2  3719  eueq3  3720  fsn  7155  bj-nuliota  37040  prprval  47439
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