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We use the SPace Infrared telescope for Cosmology and Astrophysics (SPICA) project as a template to demonstrate how deep spectrophotometric surveys covering large cosmological volumes over extended fields (1-15 deg(2)) with a mid-IR imaging spectrometer (17-36 mu m) in conjunction with deep 70 mu m photometry with a far-IR camera, at wavelengths which are not affected by dust extinction can answer the most crucial questions in current galaxy evolution studies. A SPICA-like mission will be able for the first time to provide an unobscured three-dimensional (3D, i.e. x, y, and redshift z) view of galaxy evolution back to an age of the universe of less than similar to 2 Gyrs, in the mid-IR rest frame. This survey strategy will produce a full census of the Star Formation Rate (SFR) in the universe, using polycyclic aromatic hydrocarbons (PAH) bands and fine-structure ionic lines, reaching the characteristic knee of the galaxy luminosity function, where the bulk of the population is distributed, at any redshift up to z similar to 3.5. Deep follow-up pointed spectroscopic observations with grating spectrometers onboard the satellite, across the full IR spectral range (17-210 mu m), would simultaneously measure Black Hole Accretion Rate (BHAR), from high-ionisation fine-structure lines, and SFR, from PAH and low- to mid-ionisation lines in thousands of galaxies from solar to low metallicities, down to the knee of their luminosity functions. The analysis of the resulting atlas of IR spectra will reveal the physical processes at play in evolving galaxies across cosmic time, especially its heavily dust-embedded phase during the activity peak at the cosmic noon (z similar to 1-3), through IR emission lines and features that are insensitive to the dust obscuration.