Theorem exists1 1498

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 2831.

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 1421	. 2 |- (E!x x = x <-> E.yA.x(x = x <-> x = y))
2		equid 1162	. . . . . 6 |- x = x
3	2	tbt 725	. . . . 5 |- (x = y <-> (x = y <-> x = x))
4		bicom 523	. . . . 5 |- ((x = y <-> x = x) <-> (x = x <-> x = y))
5	3, 4	bitri 171	. . . 4 |- (x = y <-> (x = x <-> x = y))
6	5	albii 1035	. . 3 |- (A.x x = y <-> A.x(x = x <-> x = y))
7	6	exbii 1087	. 2 |- (E.yA.x x = y <-> E.yA.x(x = x <-> x = y))
8		hbae 1182	. . 3 |- (A.x x = y -> A.yA.x x = y)
9	8	19.9 1072	. 2 |- (E.yA.x x = y <-> A.x x = y)
10	1, 7, 9	3bitr2i 177	1 |- (E!x x = x <-> A.x x = y)

Syntax hints:   <-> wb 144  A.wal 990   = wceq 992  E.wex 1016  E!weu 1419

This theorem is referenced by:  exists2 1499

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-10 1002  ax-12 1004  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177

This theorem depends on definitions:  df-bi 145  df-an 223  df-ex 1017  df-eu 1421

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