Implication is Left Distributive over Conjunction

Theorem

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

This can alternatively be rendered as:


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

The forms can be seen to be logically equivalent.

Proof by Natural Deduction
By the tableau method: