OK, now that you have updated your question to include the two formulas:
$$P(A\mid B) = \frac{P(A\cap B)}{P(B)} ~~ \text{provided that }
P(B) > 0, \tag{1}$$
is the definition of the conditional probability of $A$ given that
$B$ occurred. Similarly,
$$P(B\mid A) = \frac{P(B\cap A)}{P(A)} = \frac{P(A\cap B)}{P(A)} ~~ \text{provided that }
P(A) > 0, \tag{2}$$
is the definition of the conditional probability of $B$ given that
$A$ occurred. Now, it is true that it is a trivial matter to
substitute the value of $P(A\cap B)$ from $(2)$ into $(1)$ to
arrive at
$$P(A\mid B) = \frac{P(B\mid A)P(A)}{P(B)} ~~ \text{provided that }
P(A), P(B) > 0, \tag{3}$$
which is Bayes' formula but notice that Bayes's formula actually connects two different conditional probabilities $P(A\mid B)$
and $P(B\mid A)$, and is essentially a formula for "turning the
conditioning around". The Reverend Thomas Bayes referred to this
in terms of "inverse probability" and even today, there is
vigorous debate as to whether statistical inference should be
based on $P(B\mid A)$ or the inverse probability (called
the a posteriori or posterior probability).
It is undoubtedly as galling to you as it was to me when I first
discovered that Bayes' formula was just a trivial substitution of
$(2)$ into $(1)$. Perhaps if you have been born 250 years ago,
you (Note: the OP masqueraded under username AlphaBetaGamma when I wrote this answer but has since changed his username) could have made the substitution and then
people today would be talking about the AlphaBetaGamma formula and the
AlphaBetaGammian heresy and the Naive AlphaBetaGamma method$^*$ instead
of invoking Bayes' name everywhere. So
let me console you on your loss of fame by pointing out a different
version of Bayes' formula. The Law of Total Probability says
that
$$P(B) = P(B\mid A)P(A) + P(B\mid A^c)P(A^c) \tag{4}$$
and using this, we can write $(3)$ as
$$P(A\mid B) = \frac{P(B\mid A)P(A)}{P(B\mid A)P(A) + P(B\mid A^c)P(A^c)}, \tag{5}$$
or more generally as
$$P(A_i\mid B) = \frac{P(B\mid A_i)P(A_i)}{P(B\mid A_1)P(A_1) + P(B\mid A_2)P(A_2) + \cdots + P(B\mid A_n)P(A_n)}, \tag{6}$$
where the posterior probability of a possible "cause" $A_i$ of a
"datum" $B$ is related to $P(B\mid A_i)$, the likelihood of the
observation $B$ when $A_i$ is the true hypothesis and $P(A_i)$, the prior probability
(horrors!) of the hypothesis $A_i$.
$^*$ There is a famous paper R. Alpher, H. Bethe, and
G. Gamow, "The Origin of Chemical Elements", Physical Review, April 1, 1948, that is commonly referred to as
the $\alpha\beta\gamma$ paper.
AgivenB. Semantically, I'd say there's always a need to use Bayes' rule, but whenAandBare independent the rule can be reduced to a much simpler form. $\endgroup$