${\displaystyle \vdash \alpha \rightarrow \beta }$

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${\displaystyle \vdash \Box \alpha \rightarrow \Box \beta .}$

${\displaystyle \vdash \alpha \rightarrow \beta }$

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${\displaystyle \vdash \Diamond \alpha \rightarrow \Diamond \beta .}$

${\displaystyle \vdash \Box (\alpha \wedge \beta )\rightarrow \Box \alpha .}$

${\displaystyle \vdash \Box \alpha \wedge \Box \beta \rightarrow \Box (\alpha \wedge \beta )}$

${\displaystyle \vdash \Box \neg \neg \alpha \leftrightarrow \Box \alpha .}$