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We introduce a finitely axiomatizable second-order theory, which is VTC0 associated with the class FO-uniform TC0. It consists of the base theory V0 for AC0 reasoning together with the axiom NUMONES, which states the existence of a "counting array" Y for any string X: the ith row of Y contains only the number of 1 bits up to (excluding) bit i of X. We introduce the notion of "strong Δ1B-definability" for relations in a theory, and use a recursive characterization of the TC0 relations (rather than functions) to show that the TC0 relations are strongly Δ1B-definable. It follows that the TC0 functions are Σ1B-definable in VTC0. We prove a general witnessing theorem for second-order theories and conclude that theΣ1B theorems of VTC0 are witnessed by TC0 functions. We prove that VTC0 is RSUV isomorphic to the first order theory Δ1b-CR of Johannsen and Pollett (the "minimal theory for TC0"), Δ1b-CR includes the Δ1b comprehension rule, and J and P ask whether there is an upper bound to the nesting depth required for this rule. We answer "yes", because VTC0 , and therefore Δ1b-CR, are finitely axiomatizable. Finally, we show that Σ1B theorems of VTC0 translate to families of tautologies which have polynomial-size constant-depth TC0-Frege proofs. We also show that PHP is a Σ0B theorem of VTC0. These together imply that the family of propositional tautologies associated with PHP has polynomial-size constant-depth TC0-Frege proofs.