If $\alpha \vdash \phi$ and $\beta \vdash \phi$, does it mean that $\alpha \cap \beta \vdash \phi$?

Question

I would like to know if this is correct.

Answer 1

Consider the following example :

$A = \{ p \land q \}$ and $B = \{ q \land r \}$.

We have $A \vdash q$ and $B \vdash q$.

But $A \cap B = \emptyset$ and :

$\emptyset \nvdash q$,

because $q$ is not a tautology.