Theorem exists1 2549

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 4783. (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 2462	. 2 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∃𝑦∀𝑥(𝑥 = 𝑥 ↔ 𝑥 = 𝑦))
2		equid 1926	. . . . . 6 ⊢ 𝑥 = 𝑥
3	2	tbt 358	. . . . 5 ⊢ (𝑥 = 𝑦 ↔ (𝑥 = 𝑦 ↔ 𝑥 = 𝑥))
4		bicom 211	. . . . 5 ⊢ ((𝑥 = 𝑦 ↔ 𝑥 = 𝑥) ↔ (𝑥 = 𝑥 ↔ 𝑥 = 𝑦))
5	3, 4	bitri 263	. . . 4 ⊢ (𝑥 = 𝑦 ↔ (𝑥 = 𝑥 ↔ 𝑥 = 𝑦))
6	5	albii 1737	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥(𝑥 = 𝑥 ↔ 𝑥 = 𝑦))
7	6	exbii 1764	. 2 ⊢ (∃𝑦∀𝑥 𝑥 = 𝑦 ↔ ∃𝑦∀𝑥(𝑥 = 𝑥 ↔ 𝑥 = 𝑦))
8		nfae 2304	. . 3 ⊢ Ⅎ𝑦∀𝑥 𝑥 = 𝑦
9	8	19.9 2060	. 2 ⊢ (∃𝑦∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑦)
10	1, 7, 9	3bitr2i 287	1 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)

Syntax hints:   ↔ wb 195  ∀wal 1473  ∃wex 1695  ∃!weu 2458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-eu 2462

This theorem is referenced by:  exists2  2550