Three-statement biconditional

Question

If you want to show that $$p\iff q\iff r,$$ do you then have to show that $p\iff q , p\iff r , q\iff r ?$

Answer 1

Three-statement biconditional

If you want to show that $$p\iff q\iff r,\tag1$$ do you then have to show that $p\iff q , p\iff r , q\iff r ?$

I'm replying tangentially to the intent of your question. Here are three reasonable ways to read sentence $(1),$ with each main connective coloured red: $$(p\iff q)\color\red\iff r\tag A$$ $$p\color\red\iff (q\iff r)\tag B$$ $$(p\iff q)\color\red{\text{ and }}(q\iff r).\tag C$$

Readings A and B are equivalent to each other. However, neither is equivalent to the common mathematics reading, C; you can see this by supposing that $p,q,r$ are all false.

By calling sentence $(1)$ a three-statement conditional, you are almost certainly adopting reading C while observing its consequence that p⟺r (here, your three statements refer to the two conjuncts of C and p⟺r); therein lies your answer: to prove that reading C is true does not require proving that the conjunction C itself as well as some consequence of it are true. (After all, surely it is clear that proving C does not require additionally proving its consequence p⟺(q∧r) either.)