I am trying to prove a statement of the form:
If A and B, then C.
Is this equivalent to the following statement?
Given A, if B, then C.
I will interpret your question as asking whether the two statements $(A\land B)\implies C$ and $A\implies(B\implies C)$ are logically equivalent. They are.
To verify this, we can proceed slowly, by calculating the truth-table for each. Not too bad, only $8$ entries for each table. We can cut down the work by noting that if $A$ is false, each of our two sentences is easily seen to be true.
Or else we can proceed "algebraically." The sentence $(A\land B)\implies C$ is logically equivalent to $\lnot(A\land B) \lor C$, which in turn is equivalent to $(\lnot A \lor \lnot B)\lor C$.
The sentence $A\implies(B\implies C)$ is logically equivalent to $\lnot A \lor(B\implies C)$, which in turn is equivalent to $\lnot A \lor (\lnot B \lor C)$. Now we are essentially finished.
$(A \wedge B) \Rightarrow C$ means that when $A$ and $B$ are both true, so is $C$. No surprises here...
But your statement, $A \wedge (B \Rightarrow C)$ is different, since the conditional $B \Rightarrow C$ can be true when $B$ is false.