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We present a somewhat simpler variant of the doubly-efficient interactive proof systems of Goldwasser, Kalai, and Rothblum (JACM, 2015). Recall that these proof systems apply to logspace uniform sets in NC (or, more generally, to inputs that are acceptable by log-space uniform bounded-depth circuits, where the number of rounds in the proof system is linearly related to the depth of the circuit). Our simplification is in the handling of the log-space uniformity condition. Rather than having the prover provide the verifier with bits of the encoding of the circuit and establish their correctness, we employ the proof system to a highly regular universal circuit that constructs and evaluates the log-space uniform circuit in question.