If you want to remove the first items in the array for the key name Provinces and the numerical keys do not have to be preserved, you could also use array_splice:

$arr = [
    "Country" => "Canada",
    "Provinces" => [
        "Quebec",
        "Ontario",
        "British Columbia"
    ]
];
array_splice($arr["Provinces"], 0, 1);

Php demo

Or using unset to keep the numerical keys:

unset($arr['Provinces'][0]);

Use array_shift

$new_array = array_shift($array);

Try this:

function removeElementWithValue($array, $key, $value){
     foreach ($array as $subKey => $subArray) {
          if ($subArray[$key] == $value) {
               unset($array[$subKey]);
          }
     }
     return $array;
}

Then you would call it like this:

$array = removeElementWithValue($array, "year", 2011);

You can use array_shift for this:

while (($num = array_shift($arr)) !== NULL) {
  // use $num
}

In case you do not know what the first item's key is:

// Make sure to reset the array's current index
reset($array);

$key = key($array);
unset($array[$key]);

foreach ($_SESSION["addedToCart"] as &$arr)

& turns your variable into a reference instead of a copy. Normally this would be sufficient. unset() only works on data within the current scope (so your foreach loop) leaving the original unchanged (See unset() for details).

Instead you can do:

foreach ($_SESSION["addedToCart"] as $key => $val)
{
    if ($val["stoeltjes"] == $stoeltje && $val['film_id'] == $id) {
        unset($_SESSION["addedToCart"][$key]);
   }
}

The following code will do what you want:

<?php
$a = 1;
$b = 2;
$c = 3;
$d = 4;

$arr = array(
array ( $b, $d, $c, $a, $b),
array ($c, $a),
array ( $b, $d,  $c ),
array( $c, $d, $a, $b, $b)
);
echo "before:\n";
print_r($arr);

foreach($arr as $k1=>$q) {
  foreach($q as $k2=>$r) {
    if($r == $c) {
      unset($arr[$k1][$k2]);
    }
  }
}
echo "after:\n";
print_r($arr);
?>

Output:

before:
Array
(
    [0] => Array
        (
            [0] => 2
            [1] => 4
            [2] => 3
            [3] => 1
            [4] => 2
        )

    [1] => Array
        (
            [0] => 3
            [1] => 1
        )

    [2] => Array
        (
            [0] => 2
            [1] => 4
            [2] => 3
        )

    [3] => Array
        (
            [0] => 3
            [1] => 4
            [2] => 1
            [3] => 2
            [4] => 2
        )

)
after:
Array
(
    [0] => Array
        (
            [0] => 2
            [1] => 4
            [3] => 1
            [4] => 2
        )

    [1] => Array
        (
            [1] => 1
        )

    [2] => Array
        (
            [0] => 2
            [1] => 4
        )

    [3] => Array
        (
            [1] => 4
            [2] => 1
            [3] => 2
            [4] => 2
        )

)

As you can see, all the 3's have gone...