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Show that $\left(P\rightarrow Q\right) \wedge \left(Q\rightarrow R\right)$ is equivalent to $\left(P\rightarrow R\right) \wedge \left[\left(P\leftrightarrow Q\right) \vee \left(R\leftrightarrow Q\right)\right]$

After one page of expanding and collapsing the right hand side, I get as a result $$ \left(P\rightarrow Q\right) \wedge \left(Q\rightarrow P\right) \wedge \left(P\rightarrow R\right) $$

Can anybody help? Although this question has been marked as a duplicate, the linked page only provides a partial answer (one direction) and the other answer is in Polish notation which I am not familiar with.

Daniel W. Farlow
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NoBackingDown
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    Have you tried using a truth table? – Asaf Karagila Feb 20 '15 at 22:58
  • @AsafKaragila I did not, but I think the challenge of the problem is to do it without the truth table, as I already know that it should be true (hopefully). – NoBackingDown Feb 20 '15 at 23:00
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    How do you expect to get a good answer without telling us all the constraints of the problem? What can we use? – Asaf Karagila Feb 20 '15 at 23:07
  • @Dominik: take the hint and create a truth table! They provide insight that might help you realize how to proceed – daOnlyBG Feb 20 '15 at 23:13
  • Guess you are asking for a syntactic proof..what rules are you allowed to use? – Manlio Feb 20 '15 at 23:15
  • @AsafKaragila: This is a textbook problem on introductory logic and there are no contraints stated. – NoBackingDown Feb 20 '15 at 23:15
  • @Saphrosit: DeMorgan's laws, associative laws, commutative laws, absorption laws, tautology laws, contradiction laws, conditional laws, contrapositive law – NoBackingDown Feb 20 '15 at 23:19
  • If you want to do it syntactically, try writing both in terms of and, or, and not and then using laws for those. (Also, as an aside, any truth table proof should be able to be converted, perhaps at some length, to a syntactic proof.) – aes Feb 20 '15 at 23:42
  • @Dominik I posted your question from yesterday here. You may want to see it for a full solution--Tunococ's answer is quite nice. – Daniel W. Farlow Feb 22 '15 at 06:39

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