Theorem exists1 2293

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 in set.mm. (Contributed by NM, 5-Apr-2004.)

Assertion

Ref	Expression
exists1	⊢ (∃!x x = x ↔ ∀x x = y)

Distinct variable group:   x,y

Proof of Theorem exists1

Step	Hyp	Ref	Expression
1		df-eu 2208	. 2 ⊢ (∃!x x = x ↔ ∃y∀x(x = x ↔ x = y))
2		equid 1676	. . . . . 6 ⊢ x = x
3	2	tbt 333	. . . . 5 ⊢ (x = y ↔ (x = y ↔ x = x))
4		bicom 191	. . . . 5 ⊢ ((x = y ↔ x = x) ↔ (x = x ↔ x = y))
5	3, 4	bitri 240	. . . 4 ⊢ (x = y ↔ (x = x ↔ x = y))
6	5	albii 1566	. . 3 ⊢ (∀x x = y ↔ ∀x(x = x ↔ x = y))
7	6	exbii 1582	. 2 ⊢ (∃y∀x x = y ↔ ∃y∀x(x = x ↔ x = y))
8		nfae 1954	. . 3 ⊢ Ⅎy∀x x = y
9	8	19.9 1783	. 2 ⊢ (∃y∀x x = y ↔ ∀x x = y)
10	1, 7, 9	3bitr2i 264	1 ⊢ (∃!x x = x ↔ ∀x x = y)

Syntax hints:   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642  ∃!weu 2204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-eu 2208

This theorem is referenced by:  exists2  2294