# Metamath Proof Explorer

## Theorem sbiev

Description: Conversion of implicit substitution to explicit substitution. Version of sbie with a disjoint variable condition, not requiring ax-13 . See sbievw for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 30-Jun-1994) (Revised by Wolf Lammen, 18-Jan-2023) Remove dependence on ax-10 and shorten proof. (Revised by BJ, 18-Jul-2023)

Ref Expression
Hypotheses sbiev.1 ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}{\psi }$
sbiev.2 ${⊢}{x}={y}\to \left({\phi }↔{\psi }\right)$
Assertion sbiev ${⊢}\left[{y}/{x}\right]{\phi }↔{\psi }$

### Proof

Step Hyp Ref Expression
1 sbiev.1 ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}{\psi }$
2 sbiev.2 ${⊢}{x}={y}\to \left({\phi }↔{\psi }\right)$
3 sb6 ${⊢}\left[{y}/{x}\right]{\phi }↔\forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi }\right)$
4 1 2 equsalv ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi }\right)↔{\psi }$
5 3 4 bitri ${⊢}\left[{y}/{x}\right]{\phi }↔{\psi }$