Theorem eueq 2901

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 2190	. . . 4 |- ( ( x = A /\ y = A ) -> x = y )
2	1	gen2 1443	. . 3 |- A. x A. y ( ( x = A /\ y = A ) -> x = y )
3	2	biantru 300	. 2 |- ( E. x x = A <-> ( E. x x = A /\ A. x A. y ( ( x = A /\ y = A ) -> x = y ) ) )
4		isset 2736	. 2 |- ( A e. _V <-> E. x x = A )
5		eqeq1 2177	. . 3 |- ( x = y -> ( x = A <-> y = A ) )
6	5	eu4 2081	. 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 1346   = wceq 1348   E.wex 1485   E!weu 2019   e. wcel 2141   _Vcvv 2730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by:  eueq1  2902  moeq  2905  mosubt  2907  reuhypd  4456  mptfng  5323  upxp  13066