Theorem euequ1 2036

Description: Equality has existential uniqueness. (Contributed by Stefan Allan, 4-Dec-2008.)

Assertion

Ref	Expression
euequ1	|- E! x x = y

Distinct variable group:   x, y

Proof of Theorem euequ1

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		a9e 1626	. 2 |- E. x x = y
2		equtr2 1637	. . 3 |- ( ( x = y /\ z = y ) -> x = z )
3	2	gen2 1379	. 2 |- A. x A. z ( ( x = y /\ z = y ) -> x = z )
4		equequ1 1638	. . 3 |- ( x = z -> ( x = y <-> z = y ) )
5	4	eu4 2003	. 2 |- ( E! x x = y <-> ( E. x x = y /\ A. x A. z ( ( x = y /\ z = y ) -> x = z ) ) )
6	1, 3, 5	mpbir2an 883	1 |- E! x x = y

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1282   E.wex 1421   E!weu 1941

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945

This theorem is referenced by:  copsexg  3999  oprabid  5557