Check if a value exists in an array of arrays

I have an array within an array.

$array = [
    ['foo' => 'bar', 'value' => 'America'],
    ['value' => 'England', 'bar' => 'food'],
];

How do I check if 'America' exists in the array? The America array could be any key, and there could be any number of subarrays, so a generalized solution please.

Looking on the php manual, I see in_array(), but that only works for the top layer. so something like in_array("America", $a) would not work.

A general solution would be:

function deep_in_array($needle, $haystack) {
    if(in_array($needle, $haystack)) {
        return true;
    }
    foreach($haystack as $element) {
        if(is_array($element) && deep_in_array($needle, $element))
            return true;
    }
    return false;
}

The reason why I chose to use in_array and a loop is: Before I examine deeper levels of the array structure, I make sure, that the searched value is not in the current level. This way, I hope the code to be faster than doing some kind of depth-first search method.

Of course if your array is always 2 dimensional and you only want to search in this kind of arrays, then this is faster:

function in_2d_array($needle, $haystack) {
    foreach($haystack as $element) {
        if(in_array($needle, $element))
            return true;
    }
    return false;
}