For instance, to prove that $A \cup (B \cap C)=(A \cup B)\cap(A\cup C) $, can we use the definition of union and intersection, namely $A \cup B=\{x\mid x \in A \lor x \in B\}$ and $A \cap B=\{x\mid x \in A \land x \in B\}$, and derive the set or propositions $A(x)=x \text{ is in }A , B(x)=x \text{ is in } B, C(x)=x \text{ is in } C$.
Then, using logic, would it be sufficient to prove the following? $A(X) \lor (B(x) \land C(x)) \equiv (A(x) \lor B(x)) \land (A(x) \lor C(x)) $
Thanks!