When proving several statements are equivalent, are we allowed to refer to a previous proved statement? So we have three statements: (i), (ii), and (iii) and we want to prove they are equivalent. In the notes that I am reading, the author proved (i) $\implies$ (iii), (iii) $\implies$ (ii), (iii) $\implies$ (i) and finally (ii) $\implies$ (i). However, during the proof of (ii) $\implies$ (i), the author had to also use the result of (iii) $\implies$ (i). I want to ask is this allowed? I thought we were supposed to assume (ii) to prove (i) without making use of the prior results.
 A: Summarising your description: the author has shown that

*

*$A \implies C,$

*$C \implies B,$

*$C \implies A,$

*$B \implies A,$ using result #3.

That is, the author has proven that
$$(A→C)∧(C→B)∧(C→A)∧\Big((C→A)→(B→A)\Big),$$ which, via Modus Ponens, logically entails that $$(A→C)∧(C→B)∧(B→A),$$  which logically entails that $$(A↔B)∧(A↔C)∧(B↔C),$$ which means that $A,B,C$ are logically equivalent, as required.
(In fact, the two logical entailments above are actually logical equivalences, but here, this stronger assertion is unnecessary.)
Hence, the author's proof is a perfectly valid.
P.S. For interest, point #4 of my previous answer Proof of multiple equivalences contains alternative suggestions for proving that $A,B,C$ are logically equivalent.
P.P.S. It's worth pointing out that the sentence $$A↔B↔C,$$ strictly logically speaking, does not mean that $A,B,C$ are logically equivalent: it is not true when $A,B,C$ are all false. My explanation here: Associativity of logical connectives
A: Why not? Let's call the three statements A, B and C. Now the author is stating and proving a few lemmas:
Lemma 1. If A then C
Lemma 2. If C then B
Lemma 3. If C then A
Lemma 4. If B then A
In the proof of Lemma 4, the author makes use of Lemma 3, which is obviously allowed because Lemma 3 has already been proved. Now the author states
Theorem. The following conditions are equivalent:
(i) A
(ii) B
(iii) C
Proof. (i) implies (iii) is Lemma 1; (iii) implies (ii) is Lemma 2; (ii) implies (i) is Lemma 4. QED
A: Then the author's proof must have been redundant. Let's say it has already been proved that $(i) \implies (iii)$, $(iii) \implies (i)$ and $(iii) \implies (ii)$. Now to prove that all statements are equivalent, you need to prove one of the following:

*

*$(ii) \implies (iii)$

*$(ii) \implies (i)$
Now, by your claim the proof of $(ii) \implies (i)$ looks something like:
$$(ii) \implies ... \implies (iii) \implies (i)$$ Notice that stopping at $(iii)$ would have been enough.
