Prove $A \implies C$ Given, $(A \lor B) \implies C$, prove $A \implies C$
My Proof:
1 By Conditional Exchange,
$$\neg(A \lor B) \lor C$$
2 By DeMorgan's Law,
$$(\neg A \land \neg B) \lor C$$
3 By Simplification,
$$\neg A \lor C$$
4 By Conditional Exchange,
$$A \implies C$$
My question pertains to steps 2 and 3. I used Demorgan's and Simplification on a subformula of a premise -- can I do that? Usually, I would separate the subformulas, but I don't think I could do so in this case.
Thanks.

No, you need to apply Distribution before Simplification.  Simplification is not an equivalence, so cannot safely be applied to only part of a statement.
$$\begin{align}
&(A\lor B)\to C &&\text{Premise} \\ \iff & \lnot(A \lor B)\lor C&& \text{Conditional Exchange}\\\iff & (\lnot A\land\lnot B)\lor C&&\text{de Morgan's}\\\iff &(\lnot A\lor C)\land(\lnot B\lor C)&&\text{Distribution}\\\implies &\lnot A\lor C&&\text{Simplification}\\ \iff & A\to C&&\text{Conditional Exchange}
\end{align}$$
