I think 3SAT can be reduced to your problem, since

($a_1$ OR $a_2$ OR $a_3$) AND ($b_1$ OR $b_2$ OR $b_3$) AND ($c_1$ OR $c_2$ OR $c_3$) AND ...

is satisfiable iff

(NOT $A_1$ OR $a_2$ OR $a_3$) AND ($A_1$ OR $a_1$) AND (NOT $B_1$ OR $b_2$ OR $b_3$) AND ($B_1$ OR $b_1$) AND (NOT $C_1$ OR $c_2$ OR $c_3$) AND ($C_1$ OR $C_1$) AND ...

is.