Theorem eueq 2997

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 2258	. . . 4 |- ( ( x = A /\ y = A ) -> x = y )
2	1	gen2 1503	. . 3 |- A. x A. y ( ( x = A /\ y = A ) -> x = y )
3	2	biantru 302	. 2 |- ( E. x x = A <-> ( E. x x = A /\ A. x A. y ( ( x = A /\ y = A ) -> x = y ) ) )
4		isset 2828	. 2 |- ( A e. _V <-> E. x x = A )
5		eqeq1 2245	. . 3 |- ( x = y -> ( x = A <-> y = A ) )
6	5	eu4 2149	. 2 |- ( E! x x = A <-> ( E. x x = A /\ A. x A. y ( ( x = A /\ y = A ) -> x = y ) ) )
7	3, 4, 6	3bitr4i 212	1 |- ( A e. _V <-> E! x x = A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1400   = wceq 1402   E.wex 1545   E!weu 2086   e. wcel 2209   _Vcvv 2821

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-v 2823

This theorem is referenced by:  eueq1  2998  moeq  3001  mosubt  3003  reuhypd  4612  mptfng  5504  gzsum0  13690  gzsumval2  13691  upxp  15296