Theorem euequ1 2095

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 1675	. 2 |- E. x x = y
2		equtr2 1688	. . 3 |- ( ( x = y /\ z = y ) -> x = z )
3	2	gen2 1427	. 2 |- A. x A. z ( ( x = y /\ z = y ) -> x = z )
4		equequ1 1689	. . 3 |- ( x = z -> ( x = y <-> z = y ) )
5	4	eu4 2062	. 2 |- ( E! x x = y <-> ( E. x x = y /\ A. x A. z ( ( x = y /\ z = y ) -> x = z ) ) )
6	1, 3, 5	mpbir2an 927	1 |- E! x x = y

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1330   E.wex 1469   E!weu 2000

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004

This theorem is referenced by:  copsexg  4174  oprabid  5811