logic – Proving “if A or B, then C”

Suppose that I know that the following statements hold

  1. If A, then C;
  2. If B, then C.

But “if A, then C” AND “if B, then C” is logically equivalent to

  1. If A or B, then C

So I have two equivalent options when writting a proposition: state items 1 and 2 or state item 3.

Both options are proved in the same way: “Suppose A holds, then we have C… Suppose B holds, then we have C.”

Are these ideas correct?