Legal Judgment Prediction (LJP) aims to automatically predict judgment results, such as charges, relevant law articles, and the term of penalty. It plays a vital role in legal assistant systems and has become a popular research topic in recent years. This paper concerns a worthwhile but not well-studied LJP task, Appeal judgment Prediction (AJP), which predicts the judgment of an appellate court on an appeal case based on the textual description of case facts and grounds of appeal. There are two significant challenges in practice to solve the AJP task. One is how to model the appeal judgment procedure appropriately. The other is how to improve the interpretability of the prediction results. We propose a Sequential Multi-task Learning Framework with Task Dependency for Appeal Judgement Prediction (SMAJudge) to address these challenges. SMAJudge utilizes two sequential components to model the complete proceeding from the lower court to the appellate court and employs an attention mechanism to make the prediction more explainable, which handles the challenges of AJP effectively. Experimental results obtained with a dataset consisting of more than 30K appeal judgment documents have revealed the effectiveness and superiority of SMAJudge.