Theorem eueq 2974

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 2249	. . . 4 |- ( ( x = A /\ y = A ) -> x = y )
2	1	gen2 1496	. . 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 2806	. 2 |- ( A e. _V <-> E. x x = A )
5		eqeq1 2236	. . 3 |- ( x = y -> ( x = A <-> y = A ) )
6	5	eu4 2140	. 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 1393   = wceq 1395   E.wex 1538   E!weu 2077   e. wcel 2200   _Vcvv 2799

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-v 2801

This theorem is referenced by:  eueq1  2975  moeq  2978  mosubt  2980  reuhypd  4562  mptfng  5449  gsum0g  13429  gsumval2  13430  upxp  14946