TITLE: Difference between turnstile and implication
QUESTION [9 upvotes]: Does anyone know the difference between proving that
|- phi
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|- ( psi -> phi )

and proving that
|- phi -> ( psi -> phi)  ?

REPLY [3 votes]: This is not an answer to the question intended by OP, but rather a philosophical meta-answer and might be interesting to some readers.
The use of sign $\vdash$ is completely different from $\rightarrow$. You can not prove
"$\varphi \vdash \psi$", since it is not a proposition. It is a speech-act, it is invented by Frege and has been discussed extensively in philosophy of mathematics and philosophy of language. This speech-act is called assertion and is composed of two separate parts "|" and "-". It is an act of judgment. For more details see this SEP article. This distinction between a proposition and a judgment is very important and essential for Martin-Lof's type theory and philosophy.
In fact, from this viewpoint, $\varphi \vdash \psi$ is a common formal misuse because the assumptions cannot be before the speech-act. For this reason Martin-Lof prefers to write the assumptions of the assertion after the proposition in his type theories.
For difference between $\vdash$ and $\rightarrow$ see this article.