Metamath Proof Explorer


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 Gino Giotto, 10-Jan-2024) (Proof shortened by Wolf Lammen, 8-Mar-2025)

Ref Expression
Assertion cbvralsvw ( ∀ 𝑥𝐴 𝜑 ↔ ∀ 𝑦𝐴 [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step Hyp Ref Expression
1 nfv 𝑦 𝜑
2 nfs1v 𝑥 [ 𝑦 / 𝑥 ] 𝜑
3 sbequ12 ( 𝑥 = 𝑦 → ( 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
4 1 2 3 cbvralw ( ∀ 𝑥𝐴 𝜑 ↔ ∀ 𝑦𝐴 [ 𝑦 / 𝑥 ] 𝜑 )