Theorem eueq1 1913

Description: Equality has existential uniqueness.

Hypothesis

Ref	Expression
eueq1.1	|- A e. V

Assertion

Ref	Expression
eueq1	|- E!x x = A

Distinct variable group:   x,A

Proof of Theorem eueq1

Step	Hyp	Ref	Expression
1		eueq1.1	. 2 |- A e. V
2		eueq 1912	. 2 |- (A e. V <-> E!x x = A)
3	1, 2	mpbi 189	1 |- E!x x = A

Syntax hints:   = wceq 954   e. wcel 956  E!weu 1378  Vcvv 1807

This theorem is referenced by:  eueq2 1914  eueq3 1915  fnopab2 3610  fsn 3825

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808

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