Although numerous works have concentrated on minifilament eruption in complex configurations, the detailed triggering mechanism is still an open question. Using the observational data from the New Vacuum Solar Telescope and Solar Dynamics Observatory, we studied a two-step magnetic reconnection process that triggered a minifilament that erupted intermittently within a fan-spine structure in the active region NOAA 13272. The first-step reconnection occurred between a set of low-lying small-scale magnetic loops and their nearby inner spine, resulting in the appearance of a brightening at the reconnection site and the reconfiguration of the inner spine. As the reconfigured inner spine approached the outer spine, reconnection occurred between them at the null point and led to the minifilament erupting partially. Subsequently, this two-step reconnection scenario occurred again and triggered the minifilament to erupt completely. The null point reconnection was supported by the changes in the topological structure of the inner spine and the outer spine, circular ribbon flares, remote brightenings, and the brightening of the outer spine. The null point reconnection related to the second eruption was also confirmed by some plasmoids expelled from the reconnection site. Further, the results of the magnetic field extrapolation reveal the existence of a fan-spine structure involving a three-dimensional null point. We suggest that the two-step reconnection triggers the two eruptions, in which the null point reconnection plays a direct role, but the dynamical evolution of the inner spine and the outer spine driven by the first-step reconnection might be a precursor of the subsequent null point reconnection.