@Sush Maybe I should explain more in detail what I meant by: Venn diagrams can be used to show identities, not implications.

Of course, you can draw a Venn diagram and say something like. "We assume that this identity holds. If this area of the Venn diagram would be empty, then the identity would not hold. So it must be empty. Now let us have a look at this area .... "

In this way you can explain, using Venn diagram, also some results which have a form of implication.

But this is different from the most straightforward use of Venn diagrams, where you simply draw a Venn diagram and draw which areas belong to the LHS and which belong to the RHS. If the final picture is the same, this is the proof that LHS=RHS.