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I want to prove that $$\{\neg P_1\to P_2\} \vdash (\neg P_2\to P_1)$$ without using the Deduction Theorem.

I'm not sure how to proceed. The class notes are all we have to work from, no text to work on similar proofs.

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    What axioms and deduction rules are you working from?2011-04-28
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    only modus ponems2011-04-28
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    and my 3 axiom schemas which are 1. B->(A->B) 2. (~A->B)->((~A->~B)->A) 3. (A->(B->C))->((A->B)->(A->C))2011-04-28
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    Please try to put the question in the body of your message; would you require a reader of a book to look up important information on the spine?2011-04-28
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    I was asked a similar question once, and I responded by going through the proof of the deduction theorem and writing down the steps it gives for this particular case. This is a good exercise in understanding how the deduction theorem works, and technically you have not cited the deduction theorem to do it.2011-04-28
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    @QiaochuYuan So, sure this question and your comment comes as old, but there do exist systems of logic where you have the rule referenced by the author, but don't have the rule "From $\gamma$, $\alpha$ $\vdash$ $\beta$, you may infer $\gamma$, $\vdash$ C$\alpha$$\beta$." In my opinion, to best understand how the deduction theorem works, you need to write a deduction with assumptions, and then convert that deduction into a "pure" axiomatic proof using the algorithmic procedure that a meta-proof of the deduction theorem gives you.2013-08-17

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