# Hoare logic, proving conjunction rule from basic rules, possible or not?

(This is HW.) Suppose I have these following proof rules given.

I am currently considering if I can prove the conjunction rule (given below) from the above given Hoare logic rules.

My answer would be "no", because from the proof rules of Hoare logic, I can neither use the Implied rule to prove it (because from $$A$$ I cannot infer $$A \wedge B$$) nor introduce it anywhere from the first four proof rules (the only one which introduces conjunction in the post-condition requires a while). I am fairly confident that this line of reasoning is correct, but am I missing anything? For example, can I argue on a the propositional logic level that since I have $$P$$ as a precondition, and I have $$Q_1$$ and $$Q_2$$ as postconditions, I can simply introduce conjunction on the propositional logic level (instead of inferring purely on Hoare logic)?