How to prove that $A \Rightarrow (B \vee C) \equiv (A \Rightarrow B) \vee (A \Rightarrow C)$ using laws of logic

Question

This question is something that I've tried to solve on my own, but with no such luck. The only thing I've managed to do using laws of logic is using contrapositive, then DeMorgan's Laws, and other laws that lead me to $(B \wedge A) \wedge (\neg C \vee \neg A)$. I feel like I've done something wrong in my calculations and I need someone to confirm it and show me what I did wrong.

Answer 1

Using the known fact: $$ A \implies B \equiv \neg A \vee B\space ,$$

$$ \begin{aligned}A \implies (B \vee C) &\equiv \neg A \vee B \vee C \\ (A \implies B) \vee (A \implies C) &\equiv \neg A \vee B \vee \neg A \vee C \\&\equiv \neg A \vee B \vee C\end{aligned}$$