Theorem exists1 1460

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 2778.

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 1384	. 2 |- (E!x x = x <-> E.yA.x(x = x <-> x = y))
2		equid 1128	. . . . . 6 |- x = x
3	2	tbt 722	. . . . 5 |- (x = y <-> (x = y <-> x = x))
4		bicom 522	. . . . 5 |- ((x = y <-> x = x) <-> (x = x <-> x = y))
5	3, 4	bitr 173	. . . 4 |- (x = y <-> (x = x <-> x = y))
6	5	albii 1001	. . 3 |- (A.x x = y <-> A.x(x = x <-> x = y))
7	6	exbii 1053	. 2 |- (E.yA.x x = y <-> E.yA.x(x = x <-> x = y))
8		hbae 1147	. . 3 |- (A.x x = y -> A.yA.x x = y)
9	8	19.9 1038	. 2 |- (E.yA.x x = y <-> A.x x = y)
10	1, 7, 9	3bitr2 179	1 |- (E!x x = x <-> A.x x = y)

Syntax hints:   <-> wb 146  A.wal 956   = wceq 958  E.wex 982  E!weu 1382

This theorem is referenced by:  exists2 1461

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-10 968  ax-12 970  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-eu 1384

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