Theorem exists1 2037

Description: Two ways to express "only one thing exists." The left-hand side requires only one variable to express this. Both sides are false in set theory. (Contributed by NM, 5-Apr-2004.)

Assertion

Ref	Expression
exists1	|- ( E! x x = x <-> A. x x = y )

Distinct variable group:   x, y

Proof of Theorem exists1

Step	Hyp	Ref	Expression
1		df-eu 1944	. 2 |- ( E! x x = x <-> E. y A. x ( x = x <-> x = y ) )
2		equid 1629	. . . . . 6 |- x = x
3	2	tbt 245	. . . . 5 |- ( x = y <-> ( x = y <-> x = x ) )
4		bicom 138	. . . . 5 |- ( ( x = y <-> x = x ) <-> ( x = x <-> x = y ) )
5	3, 4	bitri 182	. . . 4 |- ( x = y <-> ( x = x <-> x = y ) )
6	5	albii 1399	. . 3 |- ( A. x x = y <-> A. x ( x = x <-> x = y ) )
7	6	exbii 1536	. 2 |- ( E. y A. x x = y <-> E. y A. x ( x = x <-> x = y ) )
8		hbae 1646	. . 3 |- ( A. x x = y -> A. y A. x x = y )
9	8	19.9h 1574	. 2 |- ( E. y A. x x = y <-> A. x x = y )
10	1, 7, 9	3bitr2i 206	1 |- ( E! x x = x <-> A. x x = y )

Syntax hints:   <-> wb 103   A.wal 1282   E.wex 1421   E!weu 1941

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-eu 1944

This theorem is referenced by:  exists2  2038