Metamath Proof Explorer

Theorem cbvrexvw

Description: Change the bound variable of a restricted existential quantifier using implicit substitution. Version of cbvrexv with a disjoint variable condition, which does not require ax-10 , ax-11 , ax-12 , ax-13 . (Contributed by NM, 2-Jun-1998) (Revised by Gino Giotto, 10-Jan-2024)

Ref Expression
Hypothesis cbvralvw.1 x = y φ ψ
Assertion cbvrexvw x A φ y A ψ


Step Hyp Ref Expression
1 cbvralvw.1 x = y φ ψ
2 eleq1w x = y x A y A
3 2 1 anbi12d x = y x A φ y A ψ
4 3 cbvexvw x x A φ y y A ψ
5 df-rex x A φ x x A φ
6 df-rex y A ψ y y A ψ
7 4 5 6 3bitr4i x A φ y A ψ