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In this paper, we consider a nonlinear semidefinite semi-infinite convex multiobjective optimization problem where the feasible region is determined by finite number of equality and infinite number of inequality constraints. We establish saddle point necessary and sufficient optimality conditions under some suitable constraint qualification. We establish Karush-Kuhn-Tucker optimality conditions using the saddle point optimality conditions for the differentiable case and construct some examples to illustrate our results.