Proving a chain of well-formed formulas in metalogic Suppose we have a set of assumptions $\Delta$ and can show that $\Delta \vdash A$, and then $A \vdash B$. Are we then able to say $\Delta \vdash B$? If so what lets us say this?

Generally speaking, yes. You can just concatenate the first proof and the second proof and it will yield the third proof.

Precise details of how you glue them together to get another well formed proof will depend on the deductive system. (But e.g. in a Hilbert-style system it is as simple as appending the lines of the second proof to the end of the first.)