list_pages template tag

Hi, I’m using the wp_list_pages template tag to list children pages in each parent page, and in the related children pages:

<div id="submenu">
  <?php echo $sub_pages; ?>

<?php
global $wp_query;
if( empty($wp_query->post->post_parent) ) {
$parent = $wp_query->post->ID;
} else {
$parent = $wp_query->post->post_parent;
}
wp_list_pages("title_li=&child_of=$parent");
?>

I’ve read the Codex and I know that to style it with CSS I need to use the following selectors:

.page_item { ... }
.current_page_item { ... ?

My main problem is that when using those selectors, I alter the aspect of my main navigation menu because they use the same selectors. (See https://www.ainhoavega.com and https://www.ainhoavega.com/prueba so you know what I mean…). So I wanted to know if there’s any way to alter the “submenu” without altering the main menu. I’m guessing I’ll have to assign other selectors I just don’t know how to… am I making any sense?