Drop down list dependant on IF function

I am trying to have a drop down list which is dependant on the result of a cell.

For example, if cell A1= CAT, then cell A2 should have a 'drop down list 1'; if cell A1=DOG, cell A2 would have 'drop down list 2'; if cell A1=RABBIT, cell A2 would have 'drop down list 3' etc.

Creating the drop down list is easy, I just want to know if it's possible to have different drop down lists depending on the result of a cell.