I need to prove the statement $\forall x (A(x) \to B(x)) \to (\forall xA(x) \to \forall xB(x))$ is logically valid. Here's what I'm thinking.
Assume that the statement is not logically valid. This means that the left hand side of the main implication must be true for some values while the right hand side is false for some values. Specifically $$(A(c) \to B(c)) \to (A(d) \to B(e))$$. This is where I'm stuck. If I could have instantiated all the quantifiers to c, then the right hand side of the implication would tell me A(c) would be true and B(c) would be false in order to have that whole statement be false (these correspond to A(d) and B(e)). Thus, the left hand side would also be false, which contradicts the assumption that it is true.
So am I allowed to instantiate everything to c? I don't think so, that wouldn't be general at all. So am I on the right track or completely off?
I've also tried converting to conjunctive form but that didn't go anywhere.
Best Answer
Assume the statement is false. Then the hypothesis ($\forall{x}.A(x)\rightarrow B(x))$ is true and the conclusion ($\forall{x}.A(x)\rightarrow \forall{x}.B(x))$ is false. Since the conclusion is false, its hypothesis ($\forall{x}.A(x)$) is true and its conclusion ($\forall{x}.B(x)$) is false. So there exists $c$ such that $\neg B(c)$. Specializing the other two universal quantifiers, we have $A(c)\rightarrow B(c)$ and $A(c)$, from which we deduce $B(c)$, bringing us to the desired contradiction.