Indeed, as already shown, two nested IF() functions solve your problem (See the Nested_Outcome formula in the Examples section)
I want to show another way of calculating for combined conditions, without the IF()
function.
Let's present all combinations of your conditions in the form of a table:

(I wrote "Something else" to show that for the combination of B10=FALSE AND B5=TRUE some other option is possible, and not just a space, as in your case)
Considering that FALSE is equal to 0 and TRUE is equal to 1 (at least for Google Spreadsheets this is the case), let's rewrite this table in a general form:

You can get the option number from the B5 and B10 values with a very simple formula: =B10*2+B5+1
Now use the CHOOSE() function: specify this expression as the first parameter index, and then list all four possible options separated by commas -
=CHOOSE(B10*2+B5+1,"","",C10*F6,C10*G6)
Of course, for two conditions, the benefit from this approach is not very noticeable. But the same principle can be used for both three and four conditions - it is enough to multiply the value of each condition by the next power of 2:
=CHOOSE(4th_condition*8+3rd_condition*4+2nd_condition*2+1st_condition+1,...)
Yes, the formula will be quite long. But in any case, it will be shorter and clearer than eight nested IF() functions.