# Metamath Proof Explorer

## Theorem bj-alrimg

Description: The general form of the *alrim* family of theorems: if ph is substituted for ps , then the antecedent expresses a form of nonfreeness of x in ph , so the theorem means that under a nonfreeness condition in an antecedent, one can deduce from the universally quantified implication an implication where the consequent is universally quantified. Dual of bj-exlimg . (Contributed by BJ, 9-Dec-2023)

Ref Expression
Assertion bj-alrimg ${⊢}\left({\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\psi }\right)\to \left(\forall {x}\phantom{\rule{.4em}{0ex}}\left({\psi }\to {\chi }\right)\to \left({\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\chi }\right)\right)$

### Proof

Step Hyp Ref Expression
1 sylgt ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}\left({\psi }\to {\chi }\right)\to \left(\left({\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\psi }\right)\to \left({\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\chi }\right)\right)$
2 1 com12 ${⊢}\left({\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\psi }\right)\to \left(\forall {x}\phantom{\rule{.4em}{0ex}}\left({\psi }\to {\chi }\right)\to \left({\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\chi }\right)\right)$