horizontal dry vent

I'm looking to move my washer and add a utility sink into my garage. Can I tap into the vent line (see pic). There is a sink, toilet and shower on the opposite side of the wall. Will I have venting issues? Any help would be appreciated. I talked to a plumber last week and shared this pic and...

So long story short, I am new to plumbing and am a homeowner plumbing the house I bought 1 year ago. Replumbed the whole thing myself so far, but it's a small house and fairly simple. This one thing has really got me confused though. The vent location for the clawfoot tub I'm installing is in...