$entries[0]["displayname"]="Ivan"
$entries[0]["mail"]="123@mail"
$entries[1]["displayname"]="Anna"
$entries[1]["mail"]="aaa@yandex.ru"
$entries[2]["displayname"]="Bob"
$entries[2]["mail"]="bbb@yandex.ru"
$entries[3]["displayname"]="Anna"
$entries[3]["mail"]="123@mail.ru"

$entries[0]["displayname"]="Ivan";
$entries[0]["mail"]="123@mail";
$entries[1]["displayname"]="Anna";
$entries[1]["mail"]="aaa@yandex.ru";
$entries[2]["displayname"]="Bob";
$entries[2]["mail"]="bbb@yandex.ru";
$entries[3]["displayname"]="Anna";
$entries[3]["mail"]="123@mail.ru";

// --------------------------------- //

$newArr = [];
for ($i = count($entries); $i > 0; --$i) {
    $user = $entries[$i - 1];
    $name = $user['displayname'];
    $mail = $user['mail'];

    if (isset($newArr[$name])) {
        if (strpos('@mail.ru', $mail) !== false) {
            $newArr[$name]['mail'] = $mail;
        }
    } else {
        $newArr[$name] = ['mail' => $mail];
    }
}

print_r($newArr);
// output:

Array
(
    [Anna] => Array
        (
            [mail] => 123@mail.ru
        )
    [Bob] => Array
        (
            [mail] => bbb@yandex.ru
        )
    [Ivan] => Array
        (
            [mail] => 123@mail
        )
)

<?php

$entries[0]["displayname"]="Ivan";
$entries[0]["mail"]="123@mail";
$entries[1]["displayname"]="Anna";
$entries[1]["mail"]="aaa@yandex.ru";
$entries[2]["displayname"]="Bob";
$entries[2]["mail"]="bbb@yandex.ru";
$entries[3]["displayname"]="Anna";
$entries[3]["mail"]="123@mail.ru";

$temp = [];

foreach ($entries as $item) {
    $temp[] = $item['displayname'];
}

$val = array_count_values($temp);

for ($i = 0, $j = count($entries, COUNT_RECURSIVE); $i < $j; $i++) {
    if ($val[$entries[$i]['displayname']] > 1) {
        if (strpos($entries[$i]['mail'], 'mail.ru') === false) {
            unset($entries[$i]);
        }
    }
}

echo '<pre>'; print_r($entries); echo '</pre>';
/*
Array
(
    [0] => Array
        (
            [displayname] => Ivan
            [mail] => 123@mail
        )

    [2] => Array
        (
            [displayname] => Bob
            [mail] => bbb@yandex.ru
        )

    [3] => Array
        (
            [displayname] => Anna
            [mail] => 123@mail.ru
        )

)
*/