그 사람은 선생님이다 -> 그 사람은 선생님이에요 because of '-이다', predicative postposition. '선생님이다' consists of '선생님'+'-이-'+'-다' '-이-' is stem and '-다' is ending so you can change only '-다' to '-에요' ex)'-이에요' '-이고'... '-이여' is awkward in this situation. it is used in exclamation ex) 아! 나의 젊음이여!!

~다 conjugation isn't the informal version but the declarative one; It's emotionless. You will see this kind a lot in journals, contracts, etc, but people rarely talk like this which is self explanatory. ~이에요 is the correct formal version in standard Korean. ~이여 is the informal dialect in the south west region. ~이다요 is completely wrong, but some people sometimes speak this way in Japanese accents for fun. To sum it up, conjugations are all about rules, there is no logical explanation. Take a close look on your Spanish conjugations. You won't be able to explain them all.

In the dictionary form 이다(to be), 이 is a verb stem and 다 is inifinitive ending suffix. 다 is replaced with each ending suffix. There are seven speech levels in Korean, and each level has it's ending suffix. -다 is the ending suffix of Haera-che -(어/아/여)요 and -에요 are the ending suffix Haeyo-che -어요 is used after a verb or adjective of which final vowel is ㅓ, ㅕ, ㅜ, ㅡ, ㅣ, ㅐ, ㅔ, ㅖ, ㅞ, ㅚ, ㅟ, ㅢ -아요 is used after a verb or adjective of which final vowel is ㅏ, ㅑ, ㅗ, ㅘ -여요 is used after 하다 verb -에요 is used after 이다 or 아니다 Haeyo-che(Formality: low, Politeness :high) is remarkable in that it is used both with higher level pronouns (namely, titles) as well as the middle level second person pronoun dangsin. It is used mainly: In Korean phrasebooks for foreigners. Between strangers, especially those older or of equal age. Between female co-workers or friends. Haera-che(Formality: high, Politeness :low) is generally called the "plain" style. It is used: To close friends or relatives of similar age, and by adults to children. In impersonal writing (books, newspapers, and magazines) and indirect quotations ("She said that..."). In grammar books, to give examples. For example with 그 사람은 선생님이다 그 사람은 선생님이다 is Haera-che 그 사람은 선생님이에요 is Haeyo-che with 크다 집이 크다 is Haera-che 집이 커요 is Haeyo-che with 작다 집이 작다 is Haera-che 집이 작아요 is Haeyo-che with 먹었다 사과를 먹었다 is Haera-che 사과를 먹었어요 is Haeyo-che Refer to https://en.wikipedia.org/wiki/Korean_speech_levels