Equivalence of Logical Implication and Conditional

Theorem

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

That is, the conditional is logically equivalent to logical implication.

Proof
This directly follows from:
 * The Modus Ponendo Ponens: $p \implies q, p \vdash q$;
 * The Rule of Implication: $\left({p \vdash q}\right) \vdash p \implies q$.

Caution
This is not to say that the conditional and the logical implication are the same thing.

If $p \not \vdash q$ it does not mean that $\neg \left({p \implies q}\right)$.

The latter statement is true only when $p$ is true and $q$ is false.

The former statement just says that it is not always true that when $p$ is true then $q$ is true.