Theorem exists1 2561

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; see theorem dtru 4857. (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 2474	. 2 |- ( E! x x = x <-> E. y A. x ( x = x <-> x = y ) )
2		equid 1939	. . . . . 6 |- x = x
3	2	tbt 359	. . . . 5 |- ( x = y <-> ( x = y <-> x = x ) )
4		bicom 212	. . . . 5 |- ( ( x = y <-> x = x ) <-> ( x = x <-> x = y ) )
5	3, 4	bitri 264	. . . 4 |- ( x = y <-> ( x = x <-> x = y ) )
6	5	albii 1747	. . 3 |- ( A. x x = y <-> A. x ( x = x <-> x = y ) )
7	6	exbii 1774	. 2 |- ( E. y A. x x = y <-> E. y A. x ( x = x <-> x = y ) )
8		nfae 2316	. . 3 |- F/ y A. x x = y
9	8	19.9 2072	. 2 |- ( E. y A. x x = y <-> A. x x = y )
10	1, 7, 9	3bitr2i 288	1 |- ( E! x x = x <-> A. 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  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474

This theorem is referenced by:  exists2  2562