# Iff as Conditional Chain

Originally published: 2012-12-17
Canonical URL: /2012/Dec/iff-as-conditional-chain/

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.