When does one proof of one direction of an If and Only If proof suffice?

Question

Would someone please explain when this is admissible (please expound on $\color{darkred}{sometimes}$)?

In advance of starting an Iff proof, how would one divine/previse if this convenience (of a string of equivalences) can be applied? If so, which direction should be proven manually?

Velleman, 2nd Ed, P129-130: $\color{darkred}{Sometimes}$ in a proof of a goal of the form P ↔ Q the steps in the proof of Q → P are the same as the steps used to prove P → Q, but in reverse order. In this case you may be able to simplify the proof by writing it as a string of equivalences, starting with P and ending with Q...

P130: The technique of figuring out a sequence of equivalences in one order and then writing it in the reverse order is used quite often in proofs...In particular, if you are trying to prove P ↔ Q, it is wrong to start your write-up of the proof with the unjustified statement P ↔ Q and then work out the meanings of the two sides P and Q, showing that they are the same. You should instead start with equivalences you can justify and string them together to produce a justification of the goal P ↔ Q before you assert this goal.

Answer 1

In my experience, there is no plan of making a proof bidirectional. Really, you just try to prove $P\implies Q$ and see what happens. When you have your proof that $P$ implies $Q$, you take a good look at it and try to reverse every step of it. The usual case is that you can reverse some of the steps, but not all of them, but sometimes you get lucky and can reverse every step.

However, you must always be sure that you know why a step can be reversed. For example, if you go from $x>1$ to $2x>2$, you can just as easily go from $2x>2$ to $x>1$, however, if you go from $x>1$ to $x^2>1$, you cannot reverse the step.

Answer 2

As said in the above comments, the cited statements are meaningful as "advices" about proof-strategy ... nothing is deterministic in this topic.

What is sound is the observation :

if you are trying to prove $P \equiv Q$, it is wrong to start your write-up of the proof with the unjustified statement $P \equiv Q$...

He says the (obvious but useful) advice : start with a "justified" equivalence $P' \equiv Q'$ (like an axiom of equational logic) and then work on it until you reach your goal : $P \equiv Q$.

Answer 3

There is no silver bullet, sometimes the strategy Velleman refers to works, sometimes not. However, if you try to prove $P \implies Q$ and in the course of doing so, use another statement which you know is bidirectional and non trivial, then trying to reverse the implications might be promising.

An example of what Velleman refers to might be the following:

Claim $\lambda $ is an eigenvalue of the matrix $A$ if and only if it solves the characteristic equation $det(A-\lambda I)=0$.

Proof If $\lambda$ is an eigenvalue of $A$ then there is a $x\neq0$ such that $Ax=\lambda x$, implying that $(A-\lambda I)x=0$ some $x\neq 0$, implying that $det(A-\lambda I)=0$.

Now to prove the reverse you basically just have to reverse the implications. Of course the statement that $det(A)=0$ iff $A$ is not invertible is quite non trivial. However, this statement is usually learnt before one encounters the above claim.