جَاءَ أَبِي — My father came: abī is the nominative subject, with case not overt and possessive -ī as mudāf ilayh.
رَأَيْتُ أَخِي — I saw my brother: akhī is the accusative object.
سَلَّمْتُ عَلَى أَبِي — I greeted my father: abī is genitive after ʿalā.
Do not directly transfer the overt-letter case pattern abūka / abāka / abīka to the special speaker’s-yāʾ construction.