Theorem exists1 2115

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 2022	. 2 |- ( E! x x = x <-> E. y A. x ( x = x <-> x = y ) )
2		equid 1694	. . . . . 6 |- x = x
3	2	tbt 246	. . . . 5 |- ( x = y <-> ( x = y <-> x = x ) )
4		bicom 139	. . . . 5 |- ( ( x = y <-> x = x ) <-> ( x = x <-> x = y ) )
5	3, 4	bitri 183	. . . 4 |- ( x = y <-> ( x = x <-> x = y ) )
6	5	albii 1463	. . 3 |- ( A. x x = y <-> A. x ( x = x <-> x = y ) )
7	6	exbii 1598	. 2 |- ( E. y A. x x = y <-> E. y A. x ( x = x <-> x = y ) )
8		hbae 1711	. . 3 |- ( A. x x = y -> A. y A. x x = y )
9	8	19.9h 1636	. 2 |- ( E. y A. x x = y <-> A. x x = y )
10	1, 7, 9	3bitr2i 207	1 |- ( E! x x = x <-> A. x x = y )

Syntax hints:   <-> wb 104   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-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-eu 2022

This theorem is referenced by:  exists2  2116