We propose a renormalisable model based on $A_4$ family symmetry with an $SU(5)$ grand unified theory (GUT) which leads to the minimal supersymmetric standard model (MSSM) with a two right-handed neutrino seesaw mechanism. Discrete $\mathbb{Z}9\times \mathbb{Z}_6$ symmetry provides the fermion mass hierarchy in both the quark and lepton sectors. Proton decay is highly suppressed by these symmetries. The strong CP problem is solved in a similar way to the Nelson-Barr mechanism. We discuss both the $A_4$ and $SU(5)$ symmetry breaking sectors, including doublet-triplet splitting and the origin of the $\mu$ term. The model provides an excellent fit (better than one sigma) to all quark and lepton (including neutrino) masses and mixing with spontaneous CP violation. The model predicts the entire PMNS mixing matrix with no free parameters, up to a relative phase, selected to be $2\pi/3$ from a choice of the nine complex roots of unity, which is identified as the leptogenesis phase. The model predicts a normal neutrino mass hierarchy with leptonic angles $\theta^l{13}\approx 8.7^{\circ}$, $\theta^l_{12}\approx 34^{\circ}$, $\theta^l_{23}\approx 46^{\circ}$ and an oscillation phase $\delta^l\approx -87^{\circ}$.