Solve can do this work for you. Give it the equations, tell it to solve for Pc, while eliminating what you don't want:
eqs = {Gs == HVs/HHs,
Gb == HVb/HHb,
Ps == (VVs - VHs*Gs)/(VVs + VHs*Gs),
Pb == (VVb - VHb*Gb)/(VVb + VHb*Gb),
Pc == (Ps^3 - Pb)};
Solve[eqs, Pc, {Gs, Pb, Ps, Gb}][[1, 1]] // FullSimplify
Pc -> (2 HHs HVb VHb VVs (3 HVs^2 VHs^2 + HHs^2 VVs^2) -
2 HHb HVs VHs VVb (HVs^2 VHs^2 + 3 HHs^2 VVs^2))/((HVb VHb +
HHb VVb) (HVs VHs + HHs VVs)^3)