SAT satisfaction with 10 variables
The problem you mentioned is in $P$ so it is not NP-complete. We know that $|\phi| = n$ so the number of variables is less than $n$ and we know that members of $A$ exactly have 10 True assignments. So by a brute force algorithm check every possible assignment to variables. Choose 10 variables from n, $(^{n}_{10})$ and set them to $True$ and set other variables to $False$ and check if this is a satisfying assignment. The run time of this algorithm is $(^{n}_{10})*n = O(n^{11})$.