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Default How do I use sumif and mid function in same formula

I am trying to find a solution to sum values in a column range when the value
derived by using mid function in another column range matches with the value
stores in a cell. To illustrate
A B
1 A1 10
2 A2 12
3 B1 20
4 B2 21
5
6 A 22
7 B 41

I want to total all values of range B1:B4 when value in range A1:A4 (derived
by using MID function) mataches with value in cell A6 or A7 (the first letter
of each alphanumeric value).

If this is not the correct way to arrive at the result, suggestions on
alternative way will be appreciated.
 
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