Theorem exists1 1978

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. (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 1885	. 2 ⊢ (∃!x x = x ↔ ∃y∀x(x = x ↔ x = y))
2		equid 1571	. . . . . 6 ⊢ x = x
3	2	tbt 236	. . . . 5 ⊢ (x = y ↔ (x = y ↔ x = x))
4		bicom 128	. . . . 5 ⊢ ((x = y ↔ x = x) ↔ (x = x ↔ x = y))
5	3, 4	bitri 173	. . . 4 ⊢ (x = y ↔ (x = x ↔ x = y))
6	5	albii 1339	. . 3 ⊢ (∀x x = y ↔ ∀x(x = x ↔ x = y))
7	6	exbii 1478	. 2 ⊢ (∃y∀x x = y ↔ ∃y∀x(x = x ↔ x = y))
8		hbae 1588	. . 3 ⊢ (∀x x = y → ∀y∀x x = y)
9	8	19.9h 1516	. 2 ⊢ (∃y∀x x = y ↔ ∀x x = y)
10	1, 7, 9	3bitr2i 197	1 ⊢ (∃!x x = x ↔ ∀x x = y)

Syntax hints:   ↔ wb 98  ∀wal 1226  ∃wex 1362  ∃!weu 1882

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409

This theorem depends on definitions:  df-bi 110  df-eu 1885

This theorem is referenced by:  exists2  1979