Formal proof for a Venn Diagram.

I am currently in a discrete math class (set theory), and there are problems that we have to do where we have to prove or disprove equivalency.

For Example:

Prove or disprove for all sets A , B , C that A - (B - C) = (A - C) ∪ (A ∩ B).

In class the teacher showed us how to use Venn Diagrams to informally see wether or not the statement is true, but that it does not count as a formal proof.

My question is can I somehow prove that Venn Diagrams can be used to prove or disprove set theory equivalencies and then I could use that to make Venn Diagrams a "formal proof".