Theorem exists1 2110

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 2017	. 2 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∃𝑦∀𝑥(𝑥 = 𝑥 ↔ 𝑥 = 𝑦))
2		equid 1689	. . . . . 6 ⊢ 𝑥 = 𝑥
3	2	tbt 246	. . . . 5 ⊢ (𝑥 = 𝑦 ↔ (𝑥 = 𝑦 ↔ 𝑥 = 𝑥))
4		bicom 139	. . . . 5 ⊢ ((𝑥 = 𝑦 ↔ 𝑥 = 𝑥) ↔ (𝑥 = 𝑥 ↔ 𝑥 = 𝑦))
5	3, 4	bitri 183	. . . 4 ⊢ (𝑥 = 𝑦 ↔ (𝑥 = 𝑥 ↔ 𝑥 = 𝑦))
6	5	albii 1458	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥(𝑥 = 𝑥 ↔ 𝑥 = 𝑦))
7	6	exbii 1593	. 2 ⊢ (∃𝑦∀𝑥 𝑥 = 𝑦 ↔ ∃𝑦∀𝑥(𝑥 = 𝑥 ↔ 𝑥 = 𝑦))
8		hbae 1706	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦∀𝑥 𝑥 = 𝑦)
9	8	19.9h 1631	. 2 ⊢ (∃𝑦∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑦)
10	1, 7, 9	3bitr2i 207	1 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)

Syntax hints:   ↔ wb 104  ∀wal 1341  ∃wex 1480  ∃!weu 2014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-eu 2017

This theorem is referenced by:  exists2  2111