Theorem eueq1 2713

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 2712	. 2 |- ( A e. _V <-> E! x x = A )
3	1, 2	mpbi 133	1 |- E! x x = A

Syntax hints:   = wceq 1243   e. wcel 1393   E!weu 1900   _Vcvv 2557

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559

This theorem is referenced by:  eueq2dc  2714  eueq3dc  2715  fsn  5335