Theorem euequ1 2149

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 1719	. 2 |- E. x x = y
2		equtr2 1734	. . 3 |- ( ( x = y /\ z = y ) -> x = z )
3	2	gen2 1473	. 2 |- A. x A. z ( ( x = y /\ z = y ) -> x = z )
4		equequ1 1735	. . 3 |- ( x = z -> ( x = y <-> z = y ) )
5	4	eu4 2116	. 2 |- ( E! x x = y <-> ( E. x x = y /\ A. x A. z ( ( x = y /\ z = y ) -> x = z ) ) )
6	1, 3, 5	mpbir2an 945	1 |- E! x x = y

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1371   E.wex 1515   E!weu 2054

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058

This theorem is referenced by:  copsexg  4288  oprabid  5976