To see this, we look at the contributions arising from $$\langle \Phi_H^P | = \langle \Phi_0|$$ in Eq. (1), that is we multiply with $\langle \Phi_0 |$ from the left in $$(\hat{H} -E)\sum_{P'H'}C_{H'}^{P'}|\Phi_{H'}^{P'} \rangle=0.$$