The Faddeev-Watson-Lovelace equations, which are typically used for solving three-particle scattering problems, are based on the assumption of target having one active electron while the other electrons remain passive during the collision process. So, in the case of protons scattering from helium or helium-like targets, in which there are two bound-state electrons, the passive electron has a static role in the collision channel to be studied. In this work, we intend to assign a dynamic role to all the target electrons, as they are physically active in the collision. By including an active role for the second electron in proton-helium-like collisions, a new form of the Faddeev-Watson-Lovelace integral equations is needed, in which there is no disconnected kernel. We consider the operators and the wave functions associated with the electrons to obey the Pauli exclusion principle, as the electrons are indistinguishable. In addition, a quasi-three-particle collision is assumed in the initial channel, where the electronic cloud is represented as a single identity in the collision.