  We obtain an equivalent implicit characterization of $L^p$ Banach spaces that
is amenable to a logical treatment. Using that, we obtain an axiomatization for
such spaces into a higher-order logical system, the kind of which is used in
proof mining, a research program that aims to obtain the hidden computational
content of mathematical proofs using tools from mathematical logic. As an
aside, we obtain a concrete way of formalizing $L^p$ spaces in positive-bounded
logic. The axiomatization is followed by a corresponding metatheorem in the
style of proof mining. We illustrate its use with the derivation for this class
of spaces of the standard modulus of uniform convexity.
