Theorem eueq1 1428

Description: Equality has existential uniqueness.

Hypothesis

Ref	Expression
eueq1.1	⊢ A ∈ V

Assertion

Ref	Expression
eueq1	⊢ ∃!x x = A

Distinct variable group(s):   x,A

Proof of Theorem eueq1

Step	Hyp	Ref	Expression
1		eueq1.1	. 2 ⊢ A ∈ V
2		eueq 1427	. 2 ⊢ (A ∈ V ↔ ∃!x x = A)
3	1, 2	mpbi 164	1 ⊢ ∃!x x = A

Syntax hints:  ∃!weu 1007   = wceq 1091   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  eueq2 1429  eueq3 1430  fnopab2 2747  elrnopab 2884  fsn 2895  fnoprab2 3039

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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