Theorem euequ1 2476

Description: Equality has existential uniqueness. Special case of eueq1 3379 proved using only predicate calculus. The proof needs y = z be free of x. This is ensured by having x and y be distinct. Alternately, a distinctor -. A. x x = y could have been used instead. (Contributed by Stefan Allan, 4-Dec-2008.) (Proof shortened by Wolf Lammen, 8-Sep-2019.)

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		ax6evr 1942	. . 3 |- E. z y = z
2		equequ2 1953	. . . 4 |- ( y = z -> ( x = y <-> x = z ) )
3	2	alrimiv 1855	. . 3 |- ( y = z -> A. x ( x = y <-> x = z ) )
4	1, 3	eximii 1764	. 2 |- E. z A. x ( x = y <-> x = z )
5		df-eu 2474	. 2 |- ( E! x x = y <-> E. z A. x ( x = y <-> x = z ) )
6	4, 5	mpbir 221	1 |- E! x x = y

Syntax hints:   <-> wb 196   A.wal 1481   E.wex 1704   E!weu 2470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-eu 2474

This theorem is referenced by:  copsexg  4956  oprabid  6677