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
|- ( A. x e. A ph <-> A. y e. A [ y / x ] ph )

Proof

Step Hyp Ref Expression
1 nfv
 |-  F/ y ph
2 nfs1v
 |-  F/ x [ y / x ] ph
3 sbequ12
 |-  ( x = y -> ( ph <-> [ y / x ] ph ) )
4 1 2 3 cbvralw
 |-  ( A. x e. A ph <-> A. y e. A [ y / x ] ph )