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

Ref Expression
Assertion cbvralsvw
|- ( A. x e. A ph <-> A. y e. A [ y / x ] ph )

Proof

Step Hyp Ref Expression
1 nfv
 |-  F/ z ph
2 nfs1v
 |-  F/ x [ z / x ] ph
3 sbequ12
 |-  ( x = z -> ( ph <-> [ z / x ] ph ) )
4 1 2 3 cbvralw
 |-  ( A. x e. A ph <-> A. z e. A [ z / x ] ph )
5 nfv
 |-  F/ y [ z / x ] ph
6 nfv
 |-  F/ z [ y / x ] ph
7 sbequ
 |-  ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
8 5 6 7 cbvralw
 |-  ( A. z e. A [ z / x ] ph <-> A. y e. A [ y / x ] ph )
9 4 8 bitri
 |-  ( A. x e. A ph <-> A. y e. A [ y / x ] ph )