Basement bathtub drain is roughed in close to the wall

So for those out there looking for an answer here is what my plumber did:

The toilet, sink, and shower were all fairly close to each other (in my case about 6 ft). He installed one AAV inside the cabinet under the sink. That is the only AAV installed. He said it has now become a "wet vent" to the toilet and shower drain.

Updated Answer

After watching your video, and looking at the photos, there seems to be quite a few assumptions being made, i.e. if the vent is actually a vent, what happens to the vent after it passes through the ceiling, how the toilet and bathtub are piped under the floor, and how they are actually vented.

In addition, there is also a few issues what would need to be addressed:

• There is a semi-continuous fixture draining into the assumed vent (from the water softener). It’s load would need to be accounted for when sizing the wet vent. However the 2” wet vent would be maxed out with the bathroom fixtures and the bar sink alone.
• The wye connection into the bathtub's fixture drain looks to be too close the the bathtub's trap (min should be 2x the pipe diameter, this is to protect the seal of the trap from being siphoned by the passing water when the sink drains).
• The bathtub’s trap looks like an s-trap (against code where I’m from due to s-traps having the ability to self-siphone).

Codes vary from place to place, however due to these issues and the unknown assumptions, I don't think this installation would pass a plumbing inspection, defiantly not in my area.

If I was you and I wanted to meet code, I would further investigate the vent to confirm that there are no other fixtures draining into it. I would determine what the output flow the water softener is in order to determine what it's load to the drainage and venting system would be. I would investigate the arraignment of the below grade piping to confirm how they are piped and how they are vented (this can be done via video scoping the lines). Only after this was done could I determine if code could be met.

With that being said, you didn't ask if it met code, and I've seen far worse.

Original Answer

Every fixture requires a vent. Typically washrooms are installed so that the toilet, bathtub, and basin are all vented via the basin’s vent. This is called a wet vent as the basin’s drain line is sized to allow for both the drainage of its water as well as the passage of air required to vent both the toilet and the bathtub. In addition, the fixtures served by a wet vent need to be arranged so that the toilet is either the last fixture connected or is connected via a symmetrical double wye fitting:

       Bathtub
           \
            |
            |
           /
-----------\------\
            \      \
             |     |
          Toilet   |
                 Basin

In your sketch, the bathtub is connected downstream of the toilet, which would be incorrect if the bathtub is being vented via the basin’s vent as described above. However, if the bathtub has its own individual vent then it’s not a issue.

How you pipe the new sink will depend on how the bathtub is being vented. Without knowing the arraignments and pipe sizes, I cannot give a definitive answer, but this should get you started:

If the bathtub has its own vent, and it is of sufficient size, you can simply connect the sink’s fixture arm to the vertical vent in the wall. If the bathtub is wet vented, you’ll need to connect to the main sanitary line downstream of the last fixture in the wet vent, and add a new vent from the new sink back to the existent vent stack:

     /---—---—-- Sink (c/w vent piped back to vent stack, not shown)
     |
     | Bathtub
     |     \
     |      |
     |      |
     /     /
-----------\------\
            \      \
             |     |
          Toilet   |
                 Basin