If a substance is an arrhenius base, is it necessarily a bronsted lowry acid

No! If a substance is an Arrhenius Base, is it not necessarily a Bronsted Lowery Acid.

Explanation:

Arrhenius Base produces Hydroxyl Ions [OH⁻] when dissolved in water. While, Bronsted Lowery Acid is any specie which donates proton (H⁺).

Example:

HCl + H₂O → Cl⁻ + H₃O⁺

In this reaction HCl is a Bronsted Lowery Acid. And HCl can never produce OH⁻ ions. Hence the statement is incorrect.

NaOH + H₂O → Na⁺ ₍aq₎ + OH⁻ ₍aq₎

In this reaction NaOH is ionized in water and produces OH⁻ Ions hence acting as a Arrhenius Base. And NaOH can never donate H⁺ ion so it can not act as a Bronsted Lowery Acid.