Theorem eueq1 2975

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 2974	. 2 |- ( A e. _V <-> E! x x = A )
3	1, 2	mpbi 145	1 |- E! x x = A

Syntax hints:   = wceq 1395   E!weu 2077   e. wcel 2200   _Vcvv 2799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-v 2801

This theorem is referenced by:  eueq2dc  2976  eueq3dc  2977  fsn  5807