You can simply combine multiple filters as you want. in one query statement. In the following example, example we're looking for all Pubs pubs in Cologne (inside relation 62578), which in addition are inside a given bounding box. The resulting query should look like this:
area(3600062578)->.searchArea;
node
["amenity"="pub"]
(area.searchArea)
(50.91928498040378,6.7814483642578125,51.03858753963086,6.8932342529296875);
out body;
Overpass Turbo Link: http://overpass-turbo.eu/s/ddV
If you leave out the `(area.searchArea)` line in overpass turbo, you'll notice some additional pubs, whcih are inside the bounding box, but not in the Cologne area.