That is based on the unspoken assumption that you prefer A to B. You yourself explained that such a preference is nonsense:

If my counterpart is vaccinated with A, then I'd prefer to be vaccinated with B, and vice versa. "immunizes me with vaccine A" by itself can't be assigned an utility.

If an axiom or theorem has the form "If X then Y", you should demonstrate X before invoking the axiom or theorem.