Theorem exists1 2037

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 1944	. 2 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∃𝑦∀𝑥(𝑥 = 𝑥 ↔ 𝑥 = 𝑦))
2		equid 1629	. . . . . 6 ⊢ 𝑥 = 𝑥
3	2	tbt 245	. . . . 5 ⊢ (𝑥 = 𝑦 ↔ (𝑥 = 𝑦 ↔ 𝑥 = 𝑥))
4		bicom 138	. . . . 5 ⊢ ((𝑥 = 𝑦 ↔ 𝑥 = 𝑥) ↔ (𝑥 = 𝑥 ↔ 𝑥 = 𝑦))
5	3, 4	bitri 182	. . . 4 ⊢ (𝑥 = 𝑦 ↔ (𝑥 = 𝑥 ↔ 𝑥 = 𝑦))
6	5	albii 1399	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥(𝑥 = 𝑥 ↔ 𝑥 = 𝑦))
7	6	exbii 1536	. 2 ⊢ (∃𝑦∀𝑥 𝑥 = 𝑦 ↔ ∃𝑦∀𝑥(𝑥 = 𝑥 ↔ 𝑥 = 𝑦))
8		hbae 1646	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦∀𝑥 𝑥 = 𝑦)
9	8	19.9h 1574	. 2 ⊢ (∃𝑦∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑦)
10	1, 7, 9	3bitr2i 206	1 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)

Syntax hints:   ↔ wb 103  ∀wal 1282  ∃wex 1421  ∃!weu 1941

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-eu 1944

This theorem is referenced by:  exists2  2038