Instead of binding the source collection to the list boxes, you can create a collection view source where you need filtering and bind its view to the list box.
public CollectionViewSource ViewSource1 { get; set; }
ViewSource1 = new CollectionViewSource { Source = MyList};
ViewSource1.View.Filter(predicate1)
Then bind the View of the view source to the first list box. You can bind the source collection to the second list box.
<ListBox Name="listbox1" ItemsSource="{Binding ViewSource1.View}"/>
<ListBox Name="listbox2" ItemsSource="{Binding MyList}"/>