One way might be to have the data for the chart stored in such a way that the data is 'unpivoted'. In other words, you have the X axis in one column, the series name in the next and the values (y axis) in the third. This way you can then have a slicer operating on column 2 and use it to control which series appear on the chart.
If this isn't applicable, another way, and I'm not saying this is the best way to do this, but when I had a similar problem, I duplicated the chart on a new page and used bookmarks to swap between the two. This might not be appropriate if the page has a lot of other things going on but you can copy visuals on to another page and allow them to synchronise with the originals.
