The relationship between classical functional dependency and attribute implication is investigated emphatically. Firstly, the quantifier elimination theory of Alfred Tarski is introduced. Then, the representation forms of functional dependency and the attribute implication under the quantifier elimination theory of Alfred Tarski are researched respectively, and the unified mathematics model of functional dependency and attribute implication under the Alfred Tarski algebra without quantifier is established. Then, the relationship between existences of functional dependency and attribute implication is discussed under different conversion contexts from the view of formal concept analysis, and the fundamental semantics difference between them is studied from the view of function. Finally, the problem on whether functional dependency and attribute implication can satisfy the Armsrong's axiom are researched from their satisfaction degrees according to Armsrong's axiom, the relationship between minimum dependency set and Duquenne-Guigues base is given from their satisfaction degrees according to Duquenne-Guigues base, and the relationship between classical functional dependency and attribute implication is summarized in all directions.