Theorem eueq1 2936

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 2935	. 2 |- ( A e. _V <-> E! x x = A )
3	1, 2	mpbi 145	1 |- E! x x = A

Syntax hints:   = wceq 1364   E!weu 2045   e. wcel 2167   _Vcvv 2763

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-v 2765

This theorem is referenced by:  eueq2dc  2937  eueq3dc  2938  fsn  5734