Context: GRB110709B is the first source for which Swift BAT triggered twice, with a time separation of ~10 min. The first emission (Ep. 1) goes from 40s before the 1�� trigger up to 60s after it. The second (Ep. 2) goes from 35s before the 2�� trigger to 100s after it.[...] Within the Induced Gravitational Collapse (IGC) model, we assume the progenitor to be a close binary system composed of a core of an evolved star and a Neutron Star (NS). The evolved star explodes as a Supernova (SN) and ejects material that is partially accreted by the NS. We identify this process with Ep. 1. The accretion process brings the NS over its critical mass, thus gravitationally collapsing to a BH. This process leads to the GRB emission, Ep. 2.[...] Aims: We analyze the spectra and time variability of Ep. 1 and 2 and compute the relevant parameters of the binary progenitor[...] in the IGC paradigm. Methods: We perform a time-resolved spectral analysis of Ep. 1 with a blackbody (BB) plus a power-law (PL) spectral model. We analyze Ep. 2 within the Fireshell model, identifying the Proper-GRB (P-GRB) and simulating the light curve and spectrum. We establish the redshift to be z=0.75 [...]. Results: We find for Ep. 1 a BB temperature following a broken PL with time, with the PL slopes at early and late times ��=0 and ��=-4+/-2, respectively, and a break at t=41.21s. The total energy of Ep. 1 and 2 are E_{iso}^1=1.42x10^{53}erg and E_{iso}^2=2.43x10^{52}erg, respectively. We find at transparency a Lorentz factor ��~173, laboratory radius of 6.04x10^{13}cm, P-GRB observed temperature kT_{P-GRB}=12.36keV, baryon load B=0.0057 and P-GRB energy E_{P-GRB}=3.44x10^{50}erg. [...] Conclusions: We interpret GRB110709B as a member of the IGC sources, together with GRB970828, GRB090618 and GRB101023. The XRT data during Ep. 2 offers an unprecedented tool for improving the diagnostic of GRBs emission.

12 pages, 17 figures, to appear on A&A