Theorem eueq1 3379

Description: Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.)

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 3378	. 2 |- ( A e. _V <-> E! x x = A )
3	1, 2	mpbi 220	1 |- E! x x = A

Syntax hints:   = wceq 1483   e. wcel 1990   E!weu 2470   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  eueq2  3380  eueq3  3381  fsn  6402  bj-nuliota  33019