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I was assigned a problem to do for class and i don't understand how to do it. The proof is P ↔ Q Prove: (R ↔ P) ↔ (R ↔ Q) and I am using the book language logic and Proof.

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    What have you tried so far? At which point exactly do you have problems? – lemontree Apr 30 '19 at 22:48
  • knowing how to start it. I never know how to start a bi conditional proof – Hamish Docherty Apr 30 '19 at 22:56
  • Then your first step is to take a second close look at the definitions and examples of the rules for ↔ provided in the book. If this is the book you're using (please always tell who the author is, not just the title), you will even find a section on strategies how to construct proofs for each conenctive. I'm sure you can come up with something. Getting help when you're having a problem is fine, but asking us to just do your homework for you is making things a bit too easy. – lemontree Apr 30 '19 at 23:05
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Since you say that you don't understand how to start, I will give you a hint on the skeleton and leave the details for you to fill in:

Given that the main connective in your conclusion formula is ↔, you can be relatively sure that the last rule application is a ↔I, and according to the definition of the ↔I rule (read and apply the rule definitions!), you know that the skeleton of your proof must look like this:

enter image description here

You need two subproofs, one where you derive R↔Q from R↔P, and one where you derive R↔P from R↔Q.

Since the conclusions of these subproofs are again biconditionals, the "..."'s will have the same structure - in the first subproof with R↔Q as the conclusion, derive Q from R and R from Q; the second subproof (which I leave here with ...'s) works analogously:

enter image description here

In the "..." parts of the innermost subproofs, to get from R to Q and from Q to R etc., you will need to use the premise in line 1 and the premises in line 2 or 10, respectively, together with ↔E.

With this, I'm sure you will be able to figure out the rest on your own.

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