I'm not sure I like how when we want to prove that two statements are equivalent, we typically say "A if and only if B" and we prove it by separately proving "both directions" A ⇒ B and B ⇒ A, but when we want to prove three or more statements are equivalent, we typically say "The following are equivalent" and prove a "circular chain" of conditionals (1) ⇒ (2) ⇒ [...] ⇒ (n) ⇒ (1), as if these were different proof strategies. Because really, the "both directions" business is just a special case of the chain-of-conditionals idea: (1) ⇒ (2) ⇒ (1). At the very least, one of my books ought to have mentioned this.