# Metamath Proof Explorer

## Theorem bnj1276

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj1276.1 ${⊢}{\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\phi }$
bnj1276.2 ${⊢}{\psi }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\psi }$
bnj1276.3 ${⊢}{\chi }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\chi }$
bnj1276.4 ${⊢}{\theta }↔\left({\phi }\wedge {\psi }\wedge {\chi }\right)$
Assertion bnj1276 ${⊢}{\theta }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\theta }$

### Proof

Step Hyp Ref Expression
1 bnj1276.1 ${⊢}{\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\phi }$
2 bnj1276.2 ${⊢}{\psi }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\psi }$
3 bnj1276.3 ${⊢}{\chi }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\chi }$
4 bnj1276.4 ${⊢}{\theta }↔\left({\phi }\wedge {\psi }\wedge {\chi }\right)$
5 1 2 3 hb3an ${⊢}\left({\phi }\wedge {\psi }\wedge {\chi }\right)\to \forall {x}\phantom{\rule{.4em}{0ex}}\left({\phi }\wedge {\psi }\wedge {\chi }\right)$
6 4 5 hbxfrbi ${⊢}{\theta }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\theta }$