number theory – Inequality first – why does it work?

The exercise is a following: Find all the first triplets $ (p, q, r) $ so that $ pq + qr + pr> pqr $. The solution: WLOG $ p leq q leq r $. Yes $ pq + qr + pr> pqr $then replacing $ p $ gets $ 3p ^ 2> p ^ 3 $ where we get $ p = $ 2 recital $ p $ is a prime number. Then we get an inequality 2q $ + 2r + qr> 2qr $. Replacing $ q $ we have $ q ^ 2 + 4q> 2q ^ 2 $ where we get $ q = $ 2 or $ q = $ 3. Now, substituting these values, we get that if $ q = $ 2, then all the values ​​for $ r $ give solutions and if $ q = $ 3 then either $ r = $ 3 or $ r = $ 5. The answer is correct but why? Why can you replace $ p $ and $ q $ like that? Or is it just a coincidence?