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	sbbiiev	⊢ ( [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ [ 𝑦 / 𝑥 ] ( 𝑦 ∈ 𝐴 → 𝜑 ) )
7		sbrimvw	⊢ ( [ 𝑦 / 𝑥 ] ( 𝑦 ∈ 𝐴 → 𝜑 ) ↔ ( 𝑦 ∈ 𝐴 → [ 𝑦 / 𝑥 ] 𝜑 ) )
8	6 7	bitr2i	⊢ ( ( 𝑦 ∈ 𝐴 → [ 𝑦 / 𝑥 ] 𝜑 ) ↔ [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 → 𝜑 ) )
9	8	albii	⊢ ( ∀ 𝑦 ( 𝑦 ∈ 𝐴 → [ 𝑦 / 𝑥 ] 𝜑 ) ↔ ∀ 𝑦 [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 → 𝜑 ) )
10	3 9	bitri	⊢ ( ∀ 𝑦 ∈ 𝐴 [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 → 𝜑 ) )
11	1 2 10	3bitr4i	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑦 ∈ 𝐴 [ 𝑦 / 𝑥 ] 𝜑 )