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Theorem eueq1 2707
 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 2706 . 2 (A V ↔ ∃!x x = A)
31, 2mpbi 133 1 ∃!x x = A
 Colors of variables: wff set class Syntax hints:   = wceq 1242   ∈ wcel 1390  ∃!weu 1897  Vcvv 2551 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553 This theorem is referenced by:  eueq2dc  2708  eueq3dc  2709  fsn  5278
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