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Theorem eueq1 2856
 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 2855 . 2 (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
31, 2mpbi 144 1 ∃!𝑥 𝑥 = 𝐴
 Colors of variables: wff set class Syntax hints:   = wceq 1331   ∈ wcel 1480  ∃!weu 1999  Vcvv 2686 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688 This theorem is referenced by:  eueq2dc  2857  eueq3dc  2858  fsn  5592
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