# Metamath Proof Explorer

## Theorem r19.36vf

Description: Restricted quantifier version of one direction of 19.36 . (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypothesis r19.36vf.1 ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}{\psi }$
Assertion r19.36vf ${⊢}\exists {x}\in {A}\phantom{\rule{.4em}{0ex}}\left({\phi }\to {\psi }\right)\to \left(\forall {x}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }\to {\psi }\right)$

### Proof

Step Hyp Ref Expression
1 r19.36vf.1 ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}{\psi }$
2 r19.35 ${⊢}\exists {x}\in {A}\phantom{\rule{.4em}{0ex}}\left({\phi }\to {\psi }\right)↔\left(\forall {x}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }\to \exists {x}\in {A}\phantom{\rule{.4em}{0ex}}{\psi }\right)$
3 idd ${⊢}{x}\in {A}\to \left({\psi }\to {\psi }\right)$
4 1 3 rexlimi ${⊢}\exists {x}\in {A}\phantom{\rule{.4em}{0ex}}{\psi }\to {\psi }$
5 4 imim2i ${⊢}\left(\forall {x}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }\to \exists {x}\in {A}\phantom{\rule{.4em}{0ex}}{\psi }\right)\to \left(\forall {x}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }\to {\psi }\right)$
6 2 5 sylbi ${⊢}\exists {x}\in {A}\phantom{\rule{.4em}{0ex}}\left({\phi }\to {\psi }\right)\to \left(\forall {x}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }\to {\psi }\right)$