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) (Revised by Gino Giotto, 10-Jan-2024)

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

Proof

Step	Hyp	Ref	Expression
1		nfv	⊢ Ⅎ 𝑧 𝜑
2		nfs1v	⊢ Ⅎ 𝑥 [ 𝑧 / 𝑥 ] 𝜑
3		sbequ12	⊢ ( 𝑥 = 𝑧 → ( 𝜑 ↔ [ 𝑧 / 𝑥 ] 𝜑 ) )
4	1 2 3	cbvralw	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑧 ∈ 𝐴 [ 𝑧 / 𝑥 ] 𝜑 )
5		nfv	⊢ Ⅎ 𝑦 [ 𝑧 / 𝑥 ] 𝜑
6		nfv	⊢ Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑
7		sbequ	⊢ ( 𝑧 = 𝑦 → ( [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
8	5 6 7	cbvralw	⊢ ( ∀ 𝑧 ∈ 𝐴 [ 𝑧 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 ∈ 𝐴 [ 𝑦 / 𝑥 ] 𝜑 )
9	4 8	bitri	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑦 ∈ 𝐴 [ 𝑦 / 𝑥 ] 𝜑 )