How to prove logical formulas?

Question

Mathematics

For example, an equivalence is given that expresses equivalence through other operations (the full formula is in the attachment). How can I prove such formulas?
The teacher did not accept the truth table as evidence.

Answer 1

You can put it in disjunctive normal form and compare, you can compare truth tables, or convert one to the other.

Answer 2

The most reinforced concrete and at the same time simple is to compile truth tables ;
You can also transform using identities (De Morgan, conjunction / disjunction with itself / 0/1, communicative / associative rules, etc.) - there are not so many of them, there are 15 basic ones.