For example, on proving the distributive law of set theory, the following constitutes as a proof
I am new to proof involving sets but this to me seems nothing more than replacing unions with "or" and intersections with "and", which is just a re-wording. Further more this seems to borrow some other branch of mathematics (boolean algebra specifically) entirely which is a separate development from naive set theory. It is like answering the question "why is the derivative of sin(x) equal to negative cos(x)" with some explanation from differential forms or measure theory involving the definition of differential d when limits could have sufficed - why borrow another language to explain some facts in this language?
I could have stated the law with "x is either in A or in (B and C)" is equal to "(x is in A or x is in B) and (x is in A or x is in C)", and then told to construct a proof.
Can someone enlighten me what is the importance of exchanging $\cup$ with or and $\cap$ with and and how that prove anything?