There's not a specific function for that but you can write a formula that
uses the MIN and IF functions.
This is an array formula** :
=MIN(IF(A1:A10="x",B1:B10))
=MIN(IF(A1:A10=10,B1:B10))
=MIN(IF(A1:A10=D1,B1:B10))
If a cell in column A meets the condition and the corresponding cell in
column B is empty that will evaluate as 0 and might be evaluated as the MIN
so you'd have to test for empty cells:
=MIN(IF((A1:A10="x")*(B1:B10<>""),B1:B10))
=MIN(IF((A1:A10=10)*(B1:B10<>""),B1:B10))
=MIN(IF((A1:A10=D1)*(B1:B10<>""),B1:B10))
** array formulas need to be entered using the key combination of
CTRL,SHIFT,ENTER (not just ENTER)