Theorem cbvralsvw

Description: Change bound variable by using a substitution. Version of cbvralsv with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 20-Nov-2005) Avoid ax-13 . (Revised by GG, 10-Jan-2024) (Proof shortened by Wolf Lammen, 8-Mar-2025) Avoid ax-10 , ax-12 . (Revised by SN, 21-Aug-2025)

	Ref	Expression
Assertion	cbvralsvw	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑦 ∈ 𝐴 [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sb8v	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ∀ 𝑦 [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 → 𝜑 ) )
2		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
3		df-ral	⊢ ( ∀ 𝑦 ∈ 𝐴 [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐴 → [ 𝑦 / 𝑥 ] 𝜑 ) )
4		eleq1w	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
5	4	imbi1d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ( 𝑦 ∈ 𝐴 → 𝜑 ) ) )
6	5	pm5.74i	⊢ ( ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 → 𝜑 ) ) ↔ ( 𝑥 = 𝑦 → ( 𝑦 ∈ 𝐴 → 𝜑 ) ) )
7	6	albii	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 → 𝜑 ) ) ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝑦 ∈ 𝐴 → 𝜑 ) ) )
8		sb6	⊢ ( [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 → 𝜑 ) ) )
9		sb6	⊢ ( [ 𝑦 / 𝑥 ] ( 𝑦 ∈ 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝑦 ∈ 𝐴 → 𝜑 ) ) )
10	7 8 9	3bitr4i	⊢ ( [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ [ 𝑦 / 𝑥 ] ( 𝑦 ∈ 𝐴 → 𝜑 ) )
11		sbrimvw	⊢ ( [ 𝑦 / 𝑥 ] ( 𝑦 ∈ 𝐴 → 𝜑 ) ↔ ( 𝑦 ∈ 𝐴 → [ 𝑦 / 𝑥 ] 𝜑 ) )
12	10 11	bitr2i	⊢ ( ( 𝑦 ∈ 𝐴 → [ 𝑦 / 𝑥 ] 𝜑 ) ↔ [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 → 𝜑 ) )
13	12	albii	⊢ ( ∀ 𝑦 ( 𝑦 ∈ 𝐴 → [ 𝑦 / 𝑥 ] 𝜑 ) ↔ ∀ 𝑦 [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 → 𝜑 ) )
14	3 13	bitri	⊢ ( ∀ 𝑦 ∈ 𝐴 [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 → 𝜑 ) )
15	1 2 14	3bitr4i	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑦 ∈ 𝐴 [ 𝑦 / 𝑥 ] 𝜑 )