# Conditional Constraint in MIP

I need to formulate a conditional constraint for a binary variable z defined as:

$$z_{i,j,k}$$, $$\ \ i=1:10 \ , \ j=1:5 \ , \ k=1:3$$

If any $$z_{i,j,3} = 1$$ then $$z_{i,j,1} + z_{i,j,2} = 0 \ \ \forall i,j$$

How about $$z_{i,j,1} + z_{i,j,2} \leq 2 \cdot (1 - z_{i,j,3}) \quad \forall i,j$$ ?
For simplicity, I will omit the $$i$$ and $$j$$ subscripts. Rewriting your logical proposition in conjunctive normal form somewhat automatically yields two linear constraints: $$$$z_3 \implies (\neg z_1 \land \neg z_2) \\ \neg z_3 \lor (\neg z_1 \land \neg z_2) \\ (\neg z_3 \lor \neg z_1) \land (\neg z_3 \lor \neg z_2) \\ ((1- z_3) + (1- z_1) \ge 1) \land ((1- z_3) + (1- z_2) \ge 1) \\ (z_3 + z_1 \le 1) \land (z_3 + z_2 \le 1)$$$$ Note that the big-M constraint $$z_1 + z_2 \le 2(1-z_3)$$ is weaker, being an aggregation of the previous two constraints. For example, the big-M constraint does not cut off $$z=(3/4,1/4,1/2)$$.