Min with sumproduct

Hi, I need a formula that finds the minimum from one col if criteria is met in another col, example:

A1:A4: 3,2,4,1
B1:B4: B,C,A,D

If the criteria I need is "D" and the minimum for the whole set of A1:A4 (which is 1 above) corresponds, then true.

I don't need the min only for the group D, example:

A1:A4: 1,2,4,5
B1:B4: B,D,A,D

Here the min for "D" is 2 but it is not the min for the whole set, I don't want that. I want only if "D" is matched with "1".
Thanks.