Theorem euequ1 2292

Description: Equality has existential uniqueness. Special case of eueq1 3009 proved using only predicate calculus. (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		a9ev 1656	. 2 |- E.x x = y
2		equtr2 1688	. . 3 |- ((x = y /\ z = y) -> x = z)
3	2	gen2 1547	. 2 |- A.xA.z((x = y /\ z = y) -> x = z)
4		equequ1 1684	. . 3 |- (x = z -> (x = y <-> z = y))
5	4	eu4 2243	. 2 |- (E!x x = y <-> (E.x x = y /\ A.xA.z((x = y /\ z = y) -> x = z)))
6	1, 3, 5	mpbir2an 886	1 |- E!x x = y

Syntax hints:   -> wi 4   /\ wa 358  A.wal 1540  E.wex 1541  E!weu 2204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by:  copsexg  4607  oprabid  5550  scancan  6331