# Metamath Proof Explorer

## Theorem cbv3hv

Description: Rule used to change bound variables, using implicit substitution. Version of cbv3h with a disjoint variable condition on x , y , which does not require ax-13 . Was used in a proof of axc11n (but of independent interest). (Contributed by NM, 25-Jul-2015) (Proof shortened by Wolf Lammen, 29-Nov-2020) (Proof shortened by BJ, 30-Nov-2020)

Ref Expression
Hypotheses cbv3hv.nf1 ${⊢}{\phi }\to \forall {y}\phantom{\rule{.4em}{0ex}}{\phi }$
cbv3hv.nf2 ${⊢}{\psi }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\psi }$
cbv3hv.1 ${⊢}{x}={y}\to \left({\phi }\to {\psi }\right)$
Assertion cbv3hv ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}{\phi }\to \forall {y}\phantom{\rule{.4em}{0ex}}{\psi }$

### Proof

Step Hyp Ref Expression
1 cbv3hv.nf1 ${⊢}{\phi }\to \forall {y}\phantom{\rule{.4em}{0ex}}{\phi }$
2 cbv3hv.nf2 ${⊢}{\psi }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\psi }$
3 cbv3hv.1 ${⊢}{x}={y}\to \left({\phi }\to {\psi }\right)$
4 1 nf5i ${⊢}Ⅎ{y}\phantom{\rule{.4em}{0ex}}{\phi }$
5 2 nf5i ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}{\psi }$
6 4 5 3 cbv3v ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}{\phi }\to \forall {y}\phantom{\rule{.4em}{0ex}}{\psi }$