We propose two local consistencies that extend bounds consistency (BC) by simultaneously considering com- binations of constraints as opposed to single constraints. We prove that these two local consistencies are both stronger than BC, but are NP-hard to enforce even when constraints are linear. Hence, we propose two polynomial-time techniques to enforce approximations of these two consistencies on linear constraints. One is a reformulation of the constraints on which we enforce BC whereas the other is a polynomial time algorithm. Both achieve stronger pruning than BC. Our experi- ments show large differences in favor of our approaches.