Tub/Toilet Venting

I am re-configuring my bathroom such that the tub is now going to be on a common waste line with the toilet. An existing 3" line leads to the toilet and is tied into the main stack on the other end that is 3" pipe downward and heads upward to vent out the roof.

I am planning to use a reducing wye (3" to 1.5") to tie into the line to the toilet . The wye will be about 20" from the drop at the main stack and about 24" of 1.5" piping is required between the wye and the tub trap. My question is in regards to venting. Do I need to run a vent line after my wye on the tub line to tie back into my vent stack? My inclination is that it would work fine without the vent as the 3" line will always have some air in it unless the toilet is flushed at the same time as the tub drains. Also, the tub trap is < 4 ft from the main stack that is vented already.

Thoughts??

Sketch

Here is a quick sketch:
http://picasaweb.google.com/cjtully/Bathroom#5263299906822545778.

The 3" line running horizontally hits the main stack in the wall on the right that goes up thru the roof and downward. Do I really need the vent line indicated for the tub?

vent

Still no drawing.

Link corrected...I hope

Here for sketch:

http://picasaweb.google.com/cjtully/Bathroom#5263299906822545778

Thanks

How to vent?

It sounds like a vent for the tub is well-advised. In that case, I am unsure how I can reasonably expect to vent the configuration without running a vent line horizontally.

My tub drain wye's into the 3" common drain that is shared with the toilet about 18" from the tub's trap. I assume I have to get the vent takeoff somewhere within this 18". However, this location falls about 24" from the wall where the vent line is so I cannot go vertical until that point. What am I missing?