Theorem euequ1 2114

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 1689	. 2 |- E. x x = y
2		equtr2 1704	. . 3 |- ( ( x = y /\ z = y ) -> x = z )
3	2	gen2 1443	. 2 |- A. x A. z ( ( x = y /\ z = y ) -> x = z )
4		equequ1 1705	. . 3 |- ( x = z -> ( x = y <-> z = y ) )
5	4	eu4 2081	. 2 |- ( E! x x = y <-> ( E. x x = y /\ A. x A. z ( ( x = y /\ z = y ) -> x = z ) ) )
6	1, 3, 5	mpbir2an 937	1 |- E! x x = y

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1346   E.wex 1485   E!weu 2019

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023

This theorem is referenced by:  copsexg  4229  oprabid  5885