The neutral B-meson decays into J/ψf1 and ηcf1 are analyzed phenomenologically in the perturbative QCD (PQCD) approach at the known next-to-leading order accuracy, where f1 denotes the axial-vector mesons f1(1285) and f1(1420), the mixture of flavor fn and fs states in the quark-flavor basis. With the extracted mixing angle ϕ Theo ∼ 15 • and ϕ Exp ∼ 24 • , it is of great interest to find that (a) the B → J/ψf1 and ηcf1 decays have large branching ratios(B) that are measurable at Large Hadron Collider beauty and/or Belle-II experiments in the near future. The consistent results within errors in two referenced angles indicate that much more constraints on ϕ from various aspects are still urgently demanded; (b) the large B(B 0 s → (J/ψ, ηc)f1) through f1(1285) → 2π + 2π − , f1(1285) → ηπ + π − and f1(1420) → K 0 S K + π − may be experimentally probed more easily. Several interesting ratios among the B(B → (J/ψ, ηc)f1) are presented for future exploration on the determination of |ϕ| and/or on the estimation of SU(3)F symmetry breaking; (c) the direct CP violations in the B 0 d → (J/ψ, ηc)f1 decays are generally larger than those in the corresponding B 0 s channels with a factor around 20. It suggests that the B 0 s → (J/ψ, ηc)f1 decays could provide helpful information promisingly on constraining |ϕ| due to the highly small penguin contaminations. All the results and predictions will be confronted with the future examinations.