The question: Show that a theory T is complete if and only if for any closed formulas B and C $$if \vdash_T B\ \lor C\ \Rightarrow\ \vdash_TB\ or \vdash_T C $$ Where $\vdash_T$ means that a formula P is provable in theory T.
Definition: Using definition that a theory is complete if for any formula B, either $\vdash B$ or $\vdash \lnot B$
My solution: $\Rightarrow$ If T is complete then we have either $\vdash C$ or $\vdash \lnot C$.
Now assume $\lnot (\vdash B\ or\ \vdash C)$. Then $\lnot(\vdash B)$ and $\lnot(\vdash C)$ which implies $$(\vdash \lnot B)\ \land\ (\vdash \lnot C)$$
Which means $\lnot (\vdash_T B\ \lor C)$ meaning that our hypothesis is true by modus tollens.
$\Leftarrow$ We know that $\vdash_T (B \lor \lnot B)$ because this is always true (law of excluded middle). Thus, if we have: $$\vdash_T B\ \lor C\ \Rightarrow\ \vdash_TB\ or \vdash_T C$$ for any close formula B or C, we must have that $$ \vdash_T B\ \lor \lnot B\ \Rightarrow\ \vdash_TB\ or \vdash_T \lnot B$$ And so for any B, either $\vdash B$ or $\vdash \lnot B$ meaning the theorem is complete.
Issue: I am worried as I didn't use the fact that B and C were CLOSED formula. I also would appreciate any pointers on whether my logic is correct. For example, have I used the law of excluded middle appropriately. Thanks