Proving Logical equivalence predicate formulas

Question

Proving the predicate formula below…

$\forall x((T(x)) \Rightarrow S \equiv \forall x(T(x) \Rightarrow S) $

This is my logic, may i get some suggestions on improvement or errors if i am doing anything wrong? I just started out on proving so would appreciate it if i could get any help with it.

First scenario: Suppose S is true : Suppose $\forall x ((T(x)) \Rightarrow S $ to be true, all of values in T(x) must be true, therefore, $\forall x(T(x) \Rightarrow S) $ is true as well since it only needs one value of x to be true.

Second scenario: Suppose S is true : Suppose $\forall x ((T(x)) \Rightarrow S $ to be false, it means only some cases of T(x) is true. However, if this logic applies to the $\forall x((Tx) \Rightarrow S) $, as long as some cases come true, this predicate clause would be true, which is contradictory to the first clause.

Therefore, the two formulas are not logically equivalent.

Answer 1

Correct.   The statements are not semantically equivalent, as your reasoning shows.  

Both statements will hold in any model where $S$ is true.

In models where $S$ is false, $(\forall x~T(x)) \to S$ is valid exactly when not every entity makes $T$ true.   IE: $\neg\forall x~T(x)$

In models where $S$ is false, $\forall x~(T(x)\to S)$ is valid exactly when no entity makes $T$ true.   IE: $\forall x~\neg T(x)$

Therefore $\forall x~(T(x)\to S)$ entails $(\forall x~T(x)) \to S$, but it is not entailed by that.