Theorem exists1 1072

Description: Two ways of expressing "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 1889.

Assertion

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

Distinct variable group(s):   x,y

Proof of Theorem exists1

Step	Hyp	Ref	Expression
1		df-eu 1009	. 2 |- (E!x x = x <-> E.yA.x(x = x <-> x = y))
2		eqid 810	. . . . . . . 8 |- x = x
3	2	tbt 541	. . . . . . 7 |- (x = y <-> (x = y <-> x = x))
4		bicom 398	. . . . . . 7 |- ((x = y <-> x = x) <-> (x = x <-> x = y))
5	3, 4	bitr 151	. . . . . 6 |- (x = y <-> (x = x <-> x = y))
6	5	bial 695	. . . . 5 |- (A.x x = y <-> A.x(x = x <-> x = y))
7	6	biex 733	. . . 4 |- (E.yA.x x = y <-> E.yA.x(x = x <-> x = y))
8	7	bicomi 150	. . 3 |- (E.yA.x(x = x <-> x = y) <-> E.yA.x x = y)
9		eq5 824	. . . . 5 |- (A.x x = y -> A.yA.x x = y)
10	9	19.9r 718	. . . 4 |- (A.x x = y <-> E.yA.x x = y)
11	10	bicomi 150	. . 3 |- (E.yA.x x = y <-> A.x x = y)
12	8, 11	bitr 151	. 2 |- (E.yA.x(x = x <-> x = y) <-> A.x x = y)
13	1, 12	bitr 151	1 |- (E!x x = x <-> A.x x = y)

Syntax hints:   <-> wb 127  A.wal 672  E.wex 678   = weq 797  E!weu 1007

This theorem is referenced by:  exists2 1073

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-12 802

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-eu 1009

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