We analyze Spitzer spectra of 140 active galactic nuclei (AGN) detected in the hard X-rays (14-195 keV) by the Burst Alert Telescope on board Swift. This sample allows us to probe several orders of magnitude in black hole masses (106-109 M circle dot), Eddington ratios (10-3-1), X-ray luminosities (1042-1045 erg s-1), and X-ray column densities (1020-1024 cm-2). The AGN emission is expected to be the dominant source of ionizing photons with energies greater than or similar to 50 eV, and therefore, high-ionization mid-infrared (MIR) emission lines such as [Ne v] 14.32, 24.32 mu m and [O iv] 25.89 mu m are predicted to be good proxies of AGN activity, and robust against obscuration effects. We find high detection rates (greater than or similar to 85%-90%) for the MIR coronal emission lines in our AGN sample. The luminosities of these lines are correlated with the 14-150 keV luminosity (with a typical scatter of sigma similar to 0.4-0.5 dex), strongly indicating that the MIR coronal line (CL) emission is driven by AGN activity. CLs are also tightly correlated to the bolometric luminosity (sigma similar to 0.2-0.3 dex), calculated from careful analysis of the spectral energy distribution. We find that the relationship between the CL strengths and L 14-150 keV is independent of black hole mass, AGN luminosity, and Eddington ratio, and mostly not affected by high X-ray column densities. This confirms that the MIR CLs can be used as unbiased tracers of the AGN power for X-ray luminosities in the 1042-1045 erg s-1 range.