Detecting infeasibility of System of Linear Inequalities
(1) Farkas' Lemma should show the equivalence of the infeasibility problem and LP in strongly polynomial time.

(2) Solving a linear programming problem (as a computational problem) usually means "detect the problem is either infeasible, unbounded, or having an optimal solution, and if the problem has an optimal solution, find one".  In this sense, they are equivalent.