List Grandchild Categories of Parent (but NOT Children of Parent)?

Yet another wp_list_categories question…

I am trying to list the Grandchildren of my current category, but NOT the Children. Currently this code lists all children and grandchildren categories. Anyone know can I limit this to just Grandchildren? (p.s. There is also some category-icon specific stuff in there)

<?php if (is_category()) {
  $this_category = get_category($cat);
  if (get_category_children($this_category->cat_ID) != "") {
    echo "<h1>Subcategories with Cat Icons</h1>";
    echo "<ul>";
    if (function_exists('put_cat_icons'))
    	put_cat_icons( wp_list_categories('orderby=id&show_count=0&title_li=
&use_desc_for_title=1&echo=0&child_of='.$this_category->cat_ID));
	else
		wp_list_categories('title_li');
    echo "</ul>";
  }
}?>