Bayes' theorem doesn't require much more than multiplication and division. Here's some probability definitions:

P(A) = the probability of A happening P(A|B) = the probability of A happening given B has happened P(AB) = the probability of both A and B happening

For example, if A is a fair, six-sided die rolling a 4 and B is said die rolling an even, then P(A) = 1/6, P(A|B) = 1/3, P(AB) = 1/6.

By definition, P(A|B)=P(AB)/P(B). In words, the probability of A given B is equal to the probability of both A and B divided by the probability of B.

Solving for P(AB) tells us that:

P(B)P(A|B) = P(AB) = P(A)P(B|A)

Taking out the middle and solving for P(B) allows us to flip-flop from one-side of the given to the other.

P(A|B)=P(A)*P(B|A)/P(B)

Voila! Bayes' Theorem is logically necessary.