Uniform versions of some axioms of second order arithmetic

In this paper, we discuss uniform versions of some axioms of second order arithmetic in the context of higher order arithmetic. We prove that uniform versions of weak weak König's lemma WWKL and Σ₁ ⁰ separation are equivalent to (∃²) over a suitable base theory of higher order arithmetic, where (∃²) is the assertion that there exists Φ² such that Φ f¹ = 0 if and only if ∃x⁰ (fx = 0) for all f. We also prove that uniform versions of some well-known theorems are equivalent to (∃²) or the axiom (Suslin) of the existence of the Suslin operator.