# Metamath Proof Explorer

## Theorem 19.23t

Description: Closed form of Theorem 19.23 of Margaris p. 90. See 19.23 . (Contributed by NM, 7-Nov-2005) (Proof shortened by Wolf Lammen, 13-Aug-2020) df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021) (Proof shortened by BJ, 8-Oct-2022)

Ref Expression
Assertion 19.23t ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}{\psi }\to \left(\forall {x}\phantom{\rule{.4em}{0ex}}\left({\phi }\to {\psi }\right)↔\left(\exists {x}\phantom{\rule{.4em}{0ex}}{\phi }\to {\psi }\right)\right)$

### Proof

Step Hyp Ref Expression
1 19.38b ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}{\psi }\to \left(\left(\exists {x}\phantom{\rule{.4em}{0ex}}{\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\psi }\right)↔\forall {x}\phantom{\rule{.4em}{0ex}}\left({\phi }\to {\psi }\right)\right)$
2 19.3t ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}{\psi }\to \left(\forall {x}\phantom{\rule{.4em}{0ex}}{\psi }↔{\psi }\right)$
3 2 imbi2d ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}{\psi }\to \left(\left(\exists {x}\phantom{\rule{.4em}{0ex}}{\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\psi }\right)↔\left(\exists {x}\phantom{\rule{.4em}{0ex}}{\phi }\to {\psi }\right)\right)$
4 1 3 bitr3d ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}{\psi }\to \left(\forall {x}\phantom{\rule{.4em}{0ex}}\left({\phi }\to {\psi }\right)↔\left(\exists {x}\phantom{\rule{.4em}{0ex}}{\phi }\to {\psi }\right)\right)$