What are we doing when we are proofing a theorem "if A, then B"?

Question

What are we doing when we are proofing a theorem "if A, then B"?Are we proving that "if A, then B" is true? We know "if A, then B" is true when A is true and B is true. So are we proving that B is try when A is true ? But we also know "if A, then B" is true when A is false and B is true ,or A is false and B is false. So can we prove "if A, then B" by proving B is false when A is false ? Or B is true when A is false ?

Answer 1

We are show that the statement

$$A \to B$$

is a tautology.

In other words, if $A$ is true, than there is no such a case where $B$ is false (provided $A$ is true)