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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?

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