[shell] Multiple conditions in if statement shell script

You are trying to compare strings inside an arithmetic command (((...))). Use [[ instead.

if [[ $username == "$username1" && $password == "$password1" ]] ||
   [[ $username == "$username2" && $password == "$password2" ]]; then

Note that I've reduced this to two separate tests joined by ||, with the && moved inside the tests. This is because the shell operators && and || have equal precedence and are simply evaluated from left to right. As a result, it's not generally true that a && b || c && d is equivalent to the intended ( a && b ) || ( c && d ).

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