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.