Hierarchy of operators of propositional logic

Question

I have a first order logic clause and I have to transform it to its normal clausular form. $$\forall x \exists y \left[A(x) \land \lnot B(x) \implies C(x,y) \land\exists zD(z)\right]$$

But I have several problems with the hierarchy when I have an implication in the middle of the formula.

So, I imagine these relationships as if they were of propositional logic: $a\lor b \implies c \lor d$, but now my question is:

$a\lor v \implies c \lor d$ is equal to $(a \lor b) \implies (c \lor d)$? or is equal to $a \lor (b \implies c) \lor d$?

Thanks.

Answer 1

Yes, it is right.

The formula, written with parentheses :

$∀x∃y[(A(x)∧¬B(x)) \rightarrow (C(x,y)∧∃zD(z))]$

is equivalent to :

$∀x∃y[\lnot (A(x) \land \lnot B(x)) \lor (C(x,y) \land ∃zD(z))]$

[by equivalence between $p \rightarrow q$ and $\lnot p \lor q$] which in turn is equivalent to :

$∀x∃y[(\lnot A(x) \lor B(x)) \lor (C(x,y) \land ∃zD(z))]$

[by De Morgan and double negation].