How do I test if an item is in a bash array?

Use a different kind of array: rather than an integer-indexed array, use an associative array, so the key (index) is what you will be checking for. bash-4.0 or later is required for this.

declare -A array1=( 
 [prova1]=1  [prova2]=1  [slack64]=1
)

a=slack64
[[ -n "${array1[$a]}" ]] && printf '%s is in array\n' "$a"

In the above we don't really care about the values, they need only be non-empty for this. You can "invert" an indexed array into a new associative array by exchanging the key and value:

declare -a array1=( 
 prova1 prova2 slack64
)
declare -A map    # required: declare explicit associative array
for key in "${!array1[@]}"; do map[${array1[$key]}]="$key"; done  # see below

a=slack64
[[ -n "${map[$a]}" ]] && printf '%s is in array\n' "$a"

This can pay off if you have large arrays which are frequently searched, since the implementation of associative arrays will perform better than array-traversing loops. It won't suit every use case though, since it cannot handle duplicates (though you can use the value as a counter, instead of just 1 as above), and it cannot handle an empty index.

Breaking out the complex line above, to explain the "inversion":

for key in "${!a[@]}"     # expand the array indexes to a list of words
do 
  map[${a[$key]}]="$key"  # exchange the value ${a[$key]} with the index $key
done

The straightforward way is to iterate with a loop :

var=ab
for item in "${array[@]}"; do
    [[ $var == "$item" ]] && echo "$var present in the array"
done

You can also use grep for that:

array1=(prova1 prova2 slack64)
a=slack64
if (printf '%s\n' "${array1[@]}" | grep -xq $a); then
    echo "it's in"
fi