Factor Principles

Definition

 * $$p \implies q \vdash \left({p \land r}\right) \implies \left ({q \land r}\right)$$


 * $$p \implies q \vdash \left({r \land p}\right) \implies \left ({r \land q}\right)$$

They can alternatively be rendered as:


 * $$\vdash \left({p \implies q}\right) \implies \left({\left({p \land r}\right) \implies \left ({q \land r}\right)}\right)$$


 * $$\vdash \left({p \implies q}\right) \implies \left({\left({r \land p}\right) \implies \left ({r \land q}\right)}\right)$$

The forms can be seen to be logically equivalent by application of the Rule of Implication and Modus Ponendo Ponens.

Proof by Natural Deduction
By the tableau method:

And the second is like it, namely this:

It would, of course, be possible to derive the second from the first by applying the Rule of Commutation to the conjunctions on the RHS, but this is unnecessarily fiddly for a result so obvious. The Praeclarum Theorema does all the work we need instead.