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	⊢ (∃!x x = x ↔ ∀x x = y)

Distinct variable group(s):   x,y

Proof of Theorem exists1

Step	Hyp	Ref	Expression
1		df-eu 1009	. 2 ⊢ (∃!x x = x ↔ ∃y∀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 ⊢ (∀x x = y ↔ ∀x(x = x ↔ x = y))
7	6	biex 733	. . . 4 ⊢ (∃y∀x x = y ↔ ∃y∀x(x = x ↔ x = y))
8	7	bicomi 150	. . 3 ⊢ (∃y∀x(x = x ↔ x = y) ↔ ∃y∀x x = y)
9		eq5 824	. . . . 5 ⊢ (∀x x = y → ∀y∀x x = y)
10	9	19.9r 718	. . . 4 ⊢ (∀x x = y ↔ ∃y∀x x = y)
11	10	bicomi 150	. . 3 ⊢ (∃y∀x x = y ↔ ∀x x = y)
12	8, 11	bitr 151	. 2 ⊢ (∃y∀x(x = x ↔ x = y) ↔ ∀x x = y)
13	1, 12	bitr 151	1 ⊢ (∃!x x = x ↔ ∀x x = y)

Syntax hints:   ↔ wb 127  ∀wal 672  ∃wex 678   = weq 797  ∃!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&nb‰p;674  ax-6 675  ax-7 676  ax-gsn 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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