Suppose formal system $X$ has the rule $\vdash\beta\Rightarrow\hspace{2pt}\vdash\gamma$. Is the rule $\vdash\alpha\wedge\beta\Rightarrow\hspace{2pt}\vdash\alpha\wedge\gamma$ (usually) just admissible and not derivable? How may we derive the rule $\vdash\alpha\wedge\beta\Rightarrow\hspace{2pt}\vdash\alpha\wedge\gamma$ if it is derivable?

Edit:

I now see that what I needed was $\vdash(\alpha\vee\lnot\gamma)\Rightarrow\hspace{2pt}\vdash(\alpha\vee\lnot\beta)$ on the condition that $\vdash\beta\Rightarrow\hspace{2pt}\vdash\gamma$. That would up to a point have been a meaningful question as opposed to the one I posed.I will delete this question later today.