Theorem eueq 2920

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 2207	. . . 4 |- ( ( x = A /\ y = A ) -> x = y )
2	1	gen2 1460	. . 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 2755	. 2 |- ( A e. _V <-> E. x x = A )
5		eqeq1 2194	. . 3 |- ( x = y -> ( x = A <-> y = A ) )
6	5	eu4 2098	. 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 1361   = wceq 1363   E.wex 1502   E!weu 2036   e. wcel 2158   _Vcvv 2749

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-v 2751

This theorem is referenced by:  eueq1  2921  moeq  2924  mosubt  2926  reuhypd  4483  mptfng  5353  upxp  14068