Theorem euequ1 2092

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 1674	. 2 |- E. x x = y
2		equtr2 1687	. . 3 |- ( ( x = y /\ z = y ) -> x = z )
3	2	gen2 1426	. 2 |- A. x A. z ( ( x = y /\ z = y ) -> x = z )
4		equequ1 1688	. . 3 |- ( x = z -> ( x = y <-> z = y ) )
5	4	eu4 2059	. 2 |- ( E! x x = y <-> ( E. x x = y /\ A. x A. z ( ( x = y /\ z = y ) -> x = z ) ) )
6	1, 3, 5	mpbir2an 926	1 |- E! x x = y

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1329   E.wex 1468   E!weu 1997

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001

This theorem is referenced by:  copsexg  4161  oprabid  5796