IF function using dates

Hi

I am trying to write an IF function for an escalation calculation and need some help please.
What I want to achieve is to have a date filled into cell B13 (22 February 2014). The IF function in cell F13 should look at the date filled into cell B13. If the date is equal or lower than the 15th of the month, the cell should display the previous month (January 2014). If it is higher it should display the current month in cell B13 (February 2014).
To explain it differently, dates 16 Month#1 to 15 Month#2 should deliver Month#1 as answer while dates 16 Month#2 to 15 Month#3 should deliver Month#2 as answer.

I would appreciate any assistance in this matter, thank you.

Kind Regards