By truth table, it appears to be valid.

(A ^ B) > (A > B). That is, if both A and B are true, then it's all true - truth implies truth. Otherwise the antecedent is false, When the antecedent is false, the implication is true - false implies anything.

It is valid, but care needs to be taken not to take it out of context, as that can lead to an incorrect statement.

The conclusion, stated in its most complete form (and taking the language as assumed) is:

In any theory T in which A and B are theorems, the sentence A⊃B is also a theorem.

If we move outside the context of the theory T, the consequent of that conclusion no longer holds.

So it can be counted as one of valid rules of inference (but not commonly taught in a logic class) then?

Also, does this mean that the material conditional (⊃) only reflect some, but not all elements of if-then statement in English? If A and B are unrelated to each other, but are both true, then A would still imply B?

Also, does this mean that the material conditional (⊃) only reflect some, but not all elements of if-then statement in English? If A and B are unrelated to each other, but are both true, then A would still imply B? — Varun Soontornniyomkij

"P implies Q" can be true or false. But this tells you almost nothing about either P or Q separately. (If it's false, then P is true and Q is false.) The rules and meaning of implication are so simple that even thinking about them makes them more complicated than they are, but as with many new ideas, you get used to how it works pretty quickly.

With respect to your question, spend some time with andrewk's answer. His concision makes it appear simpler than it really is. In particular, his qualifying clause, "In any theory T," is easy to overlook or forget, but it makes all the difference!