We propose a fundamental-lemma-free data-driven predictive control (DDPC) scheme for synthesizing model predictive control (MPC)-like policies directly from input-output data. Unlike the well-known DeePC approach and other DDPC methods that rely on Willems' fundamental lemma, our method avoids stacked Hankel representations and the DeePC decision variable g. Instead, we develop a closed-loop consistent, causal DDPC scheme based on the multi-step predictor Subspace-ARX (SSARX). The method first (i) estimates predictor/observer Markov parameters via a high-order ARX model to decouple the noise, then (ii) learns a multi-step past-to-future map by regression, optionally with a reduced-rank constraint. The SSARX predictor is strictly causal, which allows it to be integrated naturally into an MPC formulation. Our experimental results show that SSARX performs competitively with other methods when applied to closed-loop data affected by measurement and process noise.