display only the wp_nav_menu, which has the current-menu-item

Just intercept the nav classes and search for the current-menu-item class. This will give you the $item it’s associated with.

add_filter( 'nav_menu_css_class', 'get_active_class', 10, 2 );
function get_active_class( $classes, $item )
{
    if ( in_array( 'current-menu-item', $classes) ) {
        // We found the active class!: $item
    }

    return $classes;
}

Now just grab your nav menu and apply the needed css classes or style="display:none" to the others.