assuming A2 and B2 will always be "working days", i.e. not Saturdays or holidays then you should be able to use this formula

=B2-A2-INT((WEEKDAY(A2)+B2-A2)/7)-SUMPRODUCT((holidays>=A2)*(holidays<=B2)*(WEEKDAY(holidays)<>7))

where holidays is a named range containing your holiday dates.

If A2 or B2 might be holidays/Sats then I think you need to define how that works.....would you be excluding the start day or the end day from the count?