if events X and Y are independent, then for intersection we multiply the probability
P(Y∩X) = P(Y) * P(X)
We know that
Now we replace P(Y) * P(X) for P(Y∩X)
Cancel out P(X)
So
Like that
Now we replace P(X) * P(Y) for P(X∩Y)
Cancel out P(Y)
So
P(Y | X) = P(Y) and P(X | Y) = P(X) are true