Theorem exists1 2691

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 5423. (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 2045	. . . 4 ⊢ 𝑥 = 𝑥
2	1	bitru 1579	. . 3 ⊢ (𝑥 = 𝑥 ↔ ⊤)
3	2	eubii 2616	. 2 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∃!𝑥⊤)
4		euae 2690	. 2 ⊢ (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)
5	3, 4	bitri 278	1 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1568  ⊤wtru 1571  ∃!weu 2599

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-mo 2570  df-eu 2600

This theorem is used by:  exists2  2692