First order logic expression of "Each finite state automaton has an equivalent push-down automaton"?

Question

Problem is

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Let fsa and pda be two predicates such that fsa(x) means x is a finite state automaton and pda(y) means that y is a pushdown automaton. Let equivalent be another predicate such that equivalent(a,b) means a and b are equivalent. Which of the following first order logic statements represent the following?

      Each finite state automaton has an equivalent pushdown automaton

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I try to explain

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∀x fsa(x)→(∃y pda(y)∧equivalent(x,y))

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It is given here

(∀x fsa(x))→(∃y pda(y)∧equivalent(x,y))

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I have doubt is there typo or , is (∀x fsa(x))→(∃y pda(y)∧equivalent(x,y)) also true ?

Answer 1

In your second formula, x is not even bounded in the second part, so $$(∀x~ fsa(x))→(∃y~ pda(y)∧equivalent(x,y))$$ contains a free variable.

Even considering its universal closure, it would not mean what you want.