Filtering based on values in different columns

This feels like a simple thing,  so maybe my brain is just not on, but I could not find a quick solution - 

We have a few different Marketing programs, and I would like a slicer filter on one of my dashboards to filter down to just products from a particular category. On the item card, I have a column for each of the programs. Some items are in multiple programs, so I can't get it down to a single column to use for the slicer. I tried creating a separate table with the list of programs, but I can't have multiple relationships from that table to the Item table, or even if I create multiple intermediary tables. 

I can get around this by having a separate slicer for each program, but it would be great if I can have a single slicer that can reflect the list of different programs.

Thank you!