Metamath Proof Explorer


Theorem cbvrexsvw

Description: Change bound variable by using a substitution. Version of cbvrexsv with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 2-Mar-2008) Avoid ax-13 . (Revised by Gino Giotto, 10-Jan-2024) (Proof shortened by Wolf Lammen, 8-Mar-2025)

Ref Expression
Assertion cbvrexsvw ( ∃ 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 cbvrexw ( ∃ 𝑥𝐴 𝜑 ↔ ∃ 𝑦𝐴 [ 𝑦 / 𝑥 ] 𝜑 )