Theorem euequ1 2121

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 1696	. 2 |- E. x x = y
2		equtr2 1711	. . 3 |- ( ( x = y /\ z = y ) -> x = z )
3	2	gen2 1450	. 2 |- A. x A. z ( ( x = y /\ z = y ) -> x = z )
4		equequ1 1712	. . 3 |- ( x = z -> ( x = y <-> z = y ) )
5	4	eu4 2088	. 2 |- ( E! x x = y <-> ( E. x x = y /\ A. x A. z ( ( x = y /\ z = y ) -> x = z ) ) )
6	1, 3, 5	mpbir2an 942	1 |- E! x x = y

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1351   E.wex 1492   E!weu 2026

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030

This theorem is referenced by:  copsexg  4244  oprabid  5906