Minimum Based on a Single Criteria
The following formula returns the minimum value from B2:B10, where the corresponding value in A2:A10 equals the value in D2...
=MIN(IF(A2:A10=D2,B2:B10))
Note that the formula needs to be confirmed with CONTROL+SHIFT+ENTER. If done correctly, Excel will automatically place curly braces {...} around the formula.
Based on the sample data, the formula returns 15.
Sample Workbook: Download