Is $P((A \cap B)\cap C) = P((A \cap B) \cap (B \cap C))$ Is $P((A \cap B)\cap C) = P((A \cap B) \cap (B \cap C))$?
Drawing the Venn diagrams, the above expression seems to hold. But I am not exactly sure. Is there a more rigorous proof?

as long as you can accept the rules of set algebra manipulation
$(A \cap B) \cap (B \cap C)$
$=(A \cap (B \cap C) \cap B \cap (B \cap C)) $
$=(A \cap (B \cap C) \cap (B \cap C)) $
$=A \cap (B \cap C)  $
$=A \cap B \cap C  $
$=(A \cap B) \cap C  $
