If you're asking if you can discharge a major fixture, (your toilet) past a unvented minor one,(your washer) the answer is no.
How many fixtures in your bathroom group? What does the vent take off from. I need more details about this basement bath room before I can be more specific. Let me know the entire layout. Especially about the vent configuration. I think you're going to be able to tie your washer in to the drainage system. Just not the way you discribe. Cheers, Tom