Implication versus conjunction

Question

So I was going through the Rosen's book on Discrete Mathematics and I stumbled upon an example which had me confused from hours.

This was the question:

Consider these statements, of which the first three are premises and the fourth is a valid conclusion.
i)“All hummingbirds are richly colored.”
ii)“No large birds live on honey.”
iii)“Birds that do not live on honey are dull in color.”
iv)“Hummingbirds are small.”

Let P (x), Q(x), R(x), and S(x) be the statements “x is a hummingbird,” “x is large,” “x lives on honey,” and “x is richly colored,” respectively. Assuming that the domain consists of all birds, express the statements in the argument using quantifiers and P (x), Q(x), R(x), and S(x).

I'm confused about the second part of the problem, that is: “No large birds live on honey.” Author's answer is ¬∃x(Q(x) ∧ R(x)), but according to me ¬∃x(Q(x) → R(x)) is correct too, which is obviously not the case.
Can you tell me where I'm lacking? I've a serious confusion between implication and conjunction with quantifiers.