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

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