Theorem exists1 2693

Description: Two ways to express "exactly one thing exists". The left-hand side requires only one variable to express this. Both sides are false in set theory, see theorem dtru 5125. (Contributed by NM, 5-Apr-2004.) (Proof shortened by BJ, 7-Oct-2022.)

Assertion

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

Distinct variable group:   𝑥,𝑦

Proof of Theorem exists1

Step	Hyp	Ref	Expression
1		equid 1969	. . . 4 ⊢ 𝑥 = 𝑥
2	1	bitru 1516	. . 3 ⊢ (𝑥 = 𝑥 ↔ ⊤)
3	2	eubii 2603	. 2 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∃!𝑥⊤)
4		euae 2691	. 2 ⊢ (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)
5	3, 4	bitri 267	1 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)

Syntax hints:   ↔ wb 198  ∀wal 1505  ⊤wtru 1508  ∃!weu 2582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965

This theorem depends on definitions:  df-bi 199  df-an 388  df-tru 1510  df-ex 1743  df-mo 2547  df-eu 2583

This theorem is referenced by:  exists2  2694  exists2OLD  2695