Hello,

Remember that the converse of an implication is

The converse can be used of course to prove equivalences, since is

Here are two links talking about some uses of the converse :

Converse, Inverse, Contrapositive

Logic - Converse

If you look at the second link, you can see that there's no direct relation between the truth table of the implication and the one of the converse.

Right now, I can't remember using the converse to prove anything... Maybe I did...