Definition of Bayes Theorem

In the new “Conditional Probability” course (very nice course) we learn about the Bayes Theorem. We define this theorem as follows:

P(A | B) = P(A ∩ B) / P(B)

and

P(A ∩ B) = P(B | A) * P(A)

so after replacing of P(A ∩ B) we get the Bayes Theorem:

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

But before we learned that:

P(A ∩ B) = P(B ∩ A) -> P(A ∩ B) = P(B | A) * P(A) = P(A | B) * P(B)

so I can rewrite the Bayes theorem as follows:

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

But this solution is wrong. Where is my mistake?

Multiply both sides by P(B), and doesn’t this simply become:
P(A | B) * P(B) = P(A | B) * P(B)

Which renders this redundant because it’s not telling you anything new in that form.

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