How do I formulate a proof by equivalences in english?

Question

I have a proof of the form

Theorem. $A \iff \forall x D$.

Proof. \begin{align} A &\iff \forall x B \\ & \iff \forall x C \\ & \iff \dots \\ & \iff \forall x D \end{align} QED.

Note that $A$, $B$, $C$, etc are simply placeholders for more complex expressions/full sentences occuring in an real proof. What is the best way to write such a proof as English opposed to symbolically? This is my first attempt:

Proof. $A$ is true if and only if for all $x$ \begin{align} B, && \text{(by definition 1)} \end{align} if and only if for all $x$ \begin{align} C \end{align} if and only if for all $x$ \begin{align} \dots \end{align} if and only if for all $x$ \begin{align} D. && \text{(by lemma 3)} \end{align} QED.

The proof feels too verbose and monotonous. I also find it tiresome to repeat the clause "if and ony if for all $x$" for every step.

I considered using the following style:

Proof. $A$ is true if and only if for all $x$ \begin{align} & B, && \text{(by definition 1)} \\ &\text{if and only if}\quad C, \\ &\text{if and only if}\quad \dots \\ &\text{if and only if}\quad D. && \text{(by lemma 3)} \end{align} QED.

But I'm not sure this says the same thing, because $\textrm{false} \iff \textrm{false}$.

Is there any way to streamline my proof?

Answer 1

The point of a proof is to convince the reader that your argument is sound. To that end, streamlining should not be your goal. Verbose is OK if it's instructive. Do avoid monotony. There is no algorithm for writing a good proof. Mathematical exposition is an art.

Absent the details about $A$, $B$, $C$ and $D$ in your example I might suggest something like:

Proof:

Definition $1$ tells us that $A$ is true if and only if $B$ is true for all $x$. That is equivalent to $C$ because $\ldots$.

$\ldots$

Finally, Lemma $3$ shows that is in turn equivalent to $D$, and we're done.

Notes.

• Most proofs are sequences of implications, not logical equivalences. That allows for more varied wording.
• If you have a sequence of equivalences as in this example, you might want to state the theorem in the form "The following statements are equivalent". I once proved such a theorem, with the implication chains $A \implies B \implies C \implies D \implies A$ and $C \implies E \implies F \implies B$,
• If in your paper the statements $A$, $B$, $\ldots$ are all short and the equivalences more or less routine, then the compact exposition might be best.