Probability of getting a certain card game hand combo

Trying to figure out the chance of getting a certain combination of cards in a card game. This is related to Magic: The Gathering if that matters.

60 cards total in the deck.
Some cards are duplicates.

What is the chance of getting at least 1 of card, of which there are 4 total in the deck, and at least 1 of one or another card of which there are 4 of each, in an initial drawing of 7 cards out of the deck.

So let's say there are three groups of cards involved, cards X, Y, and Z. There are 4 of each. Drawing 7 cards out of 60 total. I need:
At least 1 X AND (at least 1 Y OR at least 1 Z)

Hopefully the question is clear enough. Thanks.