We know that we have distributive rules that handle the disjunction and the conjunction, such as the following

But, does there exist a rule, for example, that would handle distributivity of the conjunction with the implication? In other words,

At least, the above case, the answer is no. We can see this by constructing a truth table like the one below; the truth values of are not the same as . Hence, and are not logically equivalent. Therefore, it’s not a rule that we can apply in our proofs.

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