Theorem eu1

Description: An alternate way to express uniqueness used by some authors. Exercise 2(b) of Margaris p. 110. (Contributed by NM, 20-Aug-1993) (Revised by Mario Carneiro, 7-Oct-2016) (Proof shortened by Wolf Lammen, 29-Oct-2018) Avoid ax-13 . (Revised by Wolf Lammen, 7-Feb-2023)

	Ref	Expression
Hypothesis	eu1.nf	⊢ Ⅎ 𝑦 𝜑
Assertion	eu1	⊢ ( ∃! 𝑥 𝜑 ↔ ∃ 𝑥 ( 𝜑 ∧ ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑥 = 𝑦 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eu1.nf	⊢ Ⅎ 𝑦 𝜑
2		nfs1v	⊢ Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑
3	2	euf	⊢ ( ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ↔ ∃ 𝑥 ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑥 ) )
4	1	sb8euv	⊢ ( ∃! 𝑥 𝜑 ↔ ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )
5	1	sb6rfv	⊢ ( 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑥 → [ 𝑦 / 𝑥 ] 𝜑 ) )
6		equcom	⊢ ( 𝑥 = 𝑦 ↔ 𝑦 = 𝑥 )
7	6	imbi2i	⊢ ( ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑥 = 𝑦 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑦 = 𝑥 ) )
8	7	albii	⊢ ( ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑥 = 𝑦 ) ↔ ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑦 = 𝑥 ) )
9	5 8	anbi12ci	⊢ ( ( 𝜑 ∧ ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑥 = 𝑦 ) ) ↔ ( ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑦 = 𝑥 ) ∧ ∀ 𝑦 ( 𝑦 = 𝑥 → [ 𝑦 / 𝑥 ] 𝜑 ) ) )
10		albiim	⊢ ( ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑥 ) ↔ ( ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑦 = 𝑥 ) ∧ ∀ 𝑦 ( 𝑦 = 𝑥 → [ 𝑦 / 𝑥 ] 𝜑 ) ) )
11	9 10	bitr4i	⊢ ( ( 𝜑 ∧ ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑥 = 𝑦 ) ) ↔ ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑥 ) )
12	11	exbii	⊢ ( ∃ 𝑥 ( 𝜑 ∧ ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑥 = 𝑦 ) ) ↔ ∃ 𝑥 ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑥 ) )
13	3 4 12	3bitr4i	⊢ ( ∃! 𝑥 𝜑 ↔ ∃ 𝑥 ( 𝜑 ∧ ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑥 = 𝑦 ) ) )