Are $P(A,B) = P(A \text{ and } B) = P(A \cap B)$?

Question

Sorry if this is a simple question but it's important for me to ensure all of these are equal:

$$P(A,B) = P(A \text{ and } B) = P(A \cap B)$$

Also

$$P(A|B,C) = P(A|B \cap C)$$