Theorem eueq 2822

Description: Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)

Assertion

Ref	Expression
eueq	|- ( A e. _V <-> E! x x = A )

Distinct variable group:   x, A

Proof of Theorem eueq

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqtr3 2132	. . . 4 |- ( ( x = A /\ y = A ) -> x = y )
2	1	gen2 1407	. . 3 |- A. x A. y ( ( x = A /\ y = A ) -> x = y )
3	2	biantru 298	. 2 |- ( E. x x = A <-> ( E. x x = A /\ A. x A. y ( ( x = A /\ y = A ) -> x = y ) ) )
4		isset 2661	. 2 |- ( A e. _V <-> E. x x = A )
5		eqeq1 2119	. . 3 |- ( x = y -> ( x = A <-> y = A ) )
6	5	eu4 2035	. 2 |- ( E! x x = A <-> ( E. x x = A /\ A. x A. y ( ( x = A /\ y = A ) -> x = y ) ) )
7	3, 4, 6	3bitr4i 211	1 |- ( A e. _V <-> E! x x = A )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1310   = wceq 1312   E.wex 1449   e. wcel 1461   E!weu 1973   _Vcvv 2655

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-v 2657

This theorem is referenced by:  eueq1  2823  moeq  2826  mosubt  2828  reuhypd  4350  mptfng  5204  upxp  12277