Discrete Mathematics - Universal Instantiation Question from Prior Exam

Question

For my discrete one class I had the following question on an old exam:

Assume that A does not contain the free variable x 

  ∀x(A → P(x)) ≡ A → ∀x P(x)  

My professor attempted to do this on a case-by-case basis, as in what happens if A is false or if A is true and then going from there.

I am unsure how I can use that case-by-case approach to solve the problem. Any help is appreciated

Answer 1

Bi-conditional: both true or both false.

Two cases :

(i) $A$ is false.

Then $A \to Px$ is true, whatever $x$ is, and thus $\forall x \ (A \to Px)$ is true.

But if $A$ is false, $A \to \forall x Px$ is true.

(ii) $A$ is true.

Two sub-cases :

(a) $Px$ is true for all $x$. Then $A \to Px$ is true for all $x$, i.e. $\forall x \ (A \to Px)$ is true.

But if $Px$ is true for all $x$, also $\forall x Px$ is true and thus also $A \to \forall x Px$ is.

(b) $Px$ is false for some $x$. Then $A \to Px$ is false for some $x$, i.e. $\forall x \ (A \to Px)$ is false.

But if $Px$ is false for some $x$, also $\forall x Px$ is false and thus also $A \to \forall x Px$ is.

Conclusion :

$\forall x \ (A \to Px) \equiv A \to \forall x Px$

Answer 2

In my class , I 've been taught as "Law of movement of quantifiers" that $A \rightarrow \forall x P(x) \equiv \forall x (A \rightarrow P(x))$