Equivalences in Sentence Translation with Quantificational Logic

Consider the sentence "If all the birds in my house are peacocks, then Charlie is a peacock," and let the provided symbols be denoted as B for bird, H for house, P for peacock, and c for Charlie.

Are the following two translations exactly equivalent?

1. (x)( ( (Bx & Hx) → Px) → Pc)
2. ((x)( ( (Bx & Hx) → Px)) → Pc

If they are different, what is the exact difference? In particular, what is an example that works for (1) and not (2) (and vice versa). If they are equivalent, how can I generalize the situations for which the correctness of the translation is not changed by moving the position of the parentheses?

As a note, the textbook that I am using denotes (x) as a universal qualifier instead of using (∀x).

• They are not equivalent. To move quantifiers out of implications correctly see prenex normal form: (∀x ϕ) → ψ is equivalent to ∃x(ϕ → ψ), not to ∀x(ϕ → ψ). Nov 13 '19 at 18:53
• a priori that Charlie is a bird. Nov 14 '19 at 4:49
• Well 2 isn't a well-formed formula. Did you mean to add another closed parenthesis? Moreover there is an effective procedure for adding parenthesis such that these questions shouldn't arise. Nov 15 '19 at 19:18