Toilets, Sinks, Showers, Tubs and Disposals - How can I tell if cellar toilet vents OK?

I've posted here before about being less than happy with my plumbing contractor. We started out with a sealed ancient tiolet drain pipe sticking out of the cellar floor and an antique pull-chain toilet tank high on an old partition above. We gutted the whole thing and hired a plumber to install a toilet. Since the old drain indicated a previous toilet, the permit was grandfathered. Cold water was brought to the location and a toilet recently installed.

Question: how do I know if this toilet is properly vented? I've noticed churning and gurgling in this cellar toilet bowl when the third floor washing machine drains. It drains into a drain that runs from a vent on the roof straight down to a pipe in the cellar floor located about 15 feet from the toilet drain. Obviously the toilet drain connects underground to the washing machine drain. Does this mean the toilet vent is OK? Why does it churn like that when the washer drains? Thanks for any help. I'm a bit worried they've made ANOTHER mistake (previous error resulted in flooded basement and $$ settlement.)