>We start with the line integral.
Then we can make a transformation by using the following identity.
Not for the second step, the second term on the right went away because the line integral is not a function of the unprimed coordinates. That is the same reason we can pull the curl out of integral.
Now we have a form that we can use an integral theorem.
From here we can use a triple product rule for the curl and then we have to take care of psi specifically.