Theorem wl-sb8eutv

Description: Substitution of variable in universal quantifier. Closed form of sb8euv . (Contributed by Wolf Lammen, 3-May-2025)

	Ref	Expression
Assertion	wl-sb8eutv	⊢ ( ∀ 𝑥 Ⅎ 𝑦 𝜑 → ( ∃! 𝑥 𝜑 ↔ ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		nfnf1	⊢ Ⅎ 𝑦 Ⅎ 𝑦 𝜑
2	1	nfal	⊢ Ⅎ 𝑦 ∀ 𝑥 Ⅎ 𝑦 𝜑
3		equsb3	⊢ ( [ 𝑣 / 𝑥 ] 𝑥 = 𝑢 ↔ 𝑣 = 𝑢 )
4	3	sblbis	⊢ ( [ 𝑣 / 𝑥 ] ( 𝜑 ↔ 𝑥 = 𝑢 ) ↔ ( [ 𝑣 / 𝑥 ] 𝜑 ↔ 𝑣 = 𝑢 ) )
5		wl-nfsbtv	⊢ ( ∀ 𝑥 Ⅎ 𝑦 𝜑 → Ⅎ 𝑦 [ 𝑣 / 𝑥 ] 𝜑 )
6		nfvd	⊢ ( ∀ 𝑥 Ⅎ 𝑦 𝜑 → Ⅎ 𝑦 𝑣 = 𝑢 )
7	5 6	nfbid	⊢ ( ∀ 𝑥 Ⅎ 𝑦 𝜑 → Ⅎ 𝑦 ( [ 𝑣 / 𝑥 ] 𝜑 ↔ 𝑣 = 𝑢 ) )
8	4 7	nfxfrd	⊢ ( ∀ 𝑥 Ⅎ 𝑦 𝜑 → Ⅎ 𝑦 [ 𝑣 / 𝑥 ] ( 𝜑 ↔ 𝑥 = 𝑢 ) )
9		sbequ	⊢ ( 𝑣 = 𝑦 → ( [ 𝑣 / 𝑥 ] ( 𝜑 ↔ 𝑥 = 𝑢 ) ↔ [ 𝑦 / 𝑥 ] ( 𝜑 ↔ 𝑥 = 𝑢 ) ) )
10	9	a1i	⊢ ( ∀ 𝑥 Ⅎ 𝑦 𝜑 → ( 𝑣 = 𝑦 → ( [ 𝑣 / 𝑥 ] ( 𝜑 ↔ 𝑥 = 𝑢 ) ↔ [ 𝑦 / 𝑥 ] ( 𝜑 ↔ 𝑥 = 𝑢 ) ) ) )
11	2 8 10	cbvaldw	⊢ ( ∀ 𝑥 Ⅎ 𝑦 𝜑 → ( ∀ 𝑣 [ 𝑣 / 𝑥 ] ( 𝜑 ↔ 𝑥 = 𝑢 ) ↔ ∀ 𝑦 [ 𝑦 / 𝑥 ] ( 𝜑 ↔ 𝑥 = 𝑢 ) ) )
12		sb8v	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑢 ) ↔ ∀ 𝑣 [ 𝑣 / 𝑥 ] ( 𝜑 ↔ 𝑥 = 𝑢 ) )
13	12	bicomi	⊢ ( ∀ 𝑣 [ 𝑣 / 𝑥 ] ( 𝜑 ↔ 𝑥 = 𝑢 ) ↔ ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑢 ) )
14		equsb3	⊢ ( [ 𝑦 / 𝑥 ] 𝑥 = 𝑢 ↔ 𝑦 = 𝑢 )
15	14	sblbis	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 ↔ 𝑥 = 𝑢 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑢 ) )
16	15	albii	⊢ ( ∀ 𝑦 [ 𝑦 / 𝑥 ] ( 𝜑 ↔ 𝑥 = 𝑢 ) ↔ ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑢 ) )
17	11 13 16	3bitr3g	⊢ ( ∀ 𝑥 Ⅎ 𝑦 𝜑 → ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑢 ) ↔ ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑢 ) ) )
18	17	exbidv	⊢ ( ∀ 𝑥 Ⅎ 𝑦 𝜑 → ( ∃ 𝑢 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑢 ) ↔ ∃ 𝑢 ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑢 ) ) )
19		eu6	⊢ ( ∃! 𝑥 𝜑 ↔ ∃ 𝑢 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑢 ) )
20		eu6	⊢ ( ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ↔ ∃ 𝑢 ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑢 ) )
21	18 19 20	3bitr4g	⊢ ( ∀ 𝑥 Ⅎ 𝑦 𝜑 → ( ∃! 𝑥 𝜑 ↔ ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ) )