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Theorem eueq1 3009
 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 3008 . 2 (A V ↔ ∃!x x = A)
31, 2mpbi 199 1 ∃!x x = A
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Vcvv 2859 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861 This theorem is referenced by:  eueq2  3010  eueq3  3011  fnopab2  5208  fsn  5432  composefn  5818
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