We compare mid-infrared emission-line properties, from high-resolution Spitzer spectra of a hard X-ray (14 -- 195 keV) selected sample of nearby (z < 0.05) AGN detected by the Burst Alert Telescope (BAT) aboard Swift. The luminosity distribution for the mid-infrared emission-lines, [O IV] 25.89 micron, [Ne II] 12.81 micron, [Ne III] 15.56 micron and [Ne V] 14.32/24.32 micron, and hard X-ray continuum show no differences between Seyfert 1 and Seyfert 2 populations, however six newly discovered BAT AGNs are under-luminous in [O IV], most likely the result of dust extinction in the host galaxy. The overall tightness of the mid-infrared correlations and BAT fluxes and luminosities suggests that the emission lines primarily arise in gas ionized by the AGN. We also compare the mid-infrared emission-lines in the BAT AGNs with those from published studies of ULIRGs, PG QSOs, star-forming galaxies and LINERs. We find that the BAT AGN sample fall into a distinctive region when comparing the [Ne III]/[Ne II] and the [O IV]/[Ne III] ratios. These line ratios are lower in sources that have been previously classified in the mid-infrared/optical as AGN than those found for the BAT AGN, suggesting that, in our X-ray selected sample, the AGN represents the main contribution to the observed line emission. These ratios represent a new emission line diagnostic for distinguishing between AGN and star forming galaxies.
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