# Metamath Proof Explorer

## Theorem cbvex2vw

Description: Rule used to change bound variables, using implicit substitution. Version of cbvex2vv with more disjoint variable conditions, which requires fewer axioms . (Contributed by NM, 26-Jul-1995) (Revised by Gino Giotto, 10-Jan-2024)

Ref Expression
Hypothesis cbval2vw.1 ${⊢}\left({x}={z}\wedge {y}={w}\right)\to \left({\phi }↔{\psi }\right)$
Assertion cbvex2vw ${⊢}\exists {x}\phantom{\rule{.4em}{0ex}}\exists {y}\phantom{\rule{.4em}{0ex}}{\phi }↔\exists {z}\phantom{\rule{.4em}{0ex}}\exists {w}\phantom{\rule{.4em}{0ex}}{\psi }$

### Proof

Step Hyp Ref Expression
1 cbval2vw.1 ${⊢}\left({x}={z}\wedge {y}={w}\right)\to \left({\phi }↔{\psi }\right)$
2 1 cbvexdvaw ${⊢}{x}={z}\to \left(\exists {y}\phantom{\rule{.4em}{0ex}}{\phi }↔\exists {w}\phantom{\rule{.4em}{0ex}}{\psi }\right)$
3 2 cbvexvw ${⊢}\exists {x}\phantom{\rule{.4em}{0ex}}\exists {y}\phantom{\rule{.4em}{0ex}}{\phi }↔\exists {z}\phantom{\rule{.4em}{0ex}}\exists {w}\phantom{\rule{.4em}{0ex}}{\psi }$