Implication is Left Distributive over Conjunction

Theorem

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

This can alternatively be rendered as:


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

The forms can be seen to be logically equivalent.

Proof by Natural Deduction
By the tableau method: