Answer
See step by step work for proof.
Work Step by Step
To prove- $A \cap (B \cap C)=(A \cap B) \cap C$
Proof-
$A \cap (B \cap C)$={$x|x \in A \cap (B \cap C)$
Using the definition of intersection, an element of $A \cap (B \cap C)$ is an element that is in $A$ and in $B \cap C$
={$x|x \in A \land x \in B \cap C$}
Using the definition of intersection, an element of $B \cap C$ is an element that is in $B$ and in $C$.
={$x|x \in A \land (x \in B \land x \in C)$}
Using the associative law for propositions:
={$x|(x \in A \land x \in B) \land x \in C$}
Use the definition of intersection:
={$x|x \in (A \cap B) \land x \in C$}
Use the definition of intersection again:
={$x|x \in (A \cap B) \cap C$}
=$(A \cap B) \cap C$
Thus, we have shown that $(A \cap B) \cap C$=$A \cap (B \cap C)$
