Yes, but with an important condition: the variable "x" must be "fresh", i.e., not used in the current derivation before. Consider two arguments.It says that if you have a formula like:
for all x : (f x)
then you can prove it by simply proving:
1. "Every man has two parents. Indeed, take any man; let's call him Joe. Then ... " This is the beginning of a proper logical argument.
2. "Every man is a drunkard. Indeed, take our neighbor Joe. ..." This is more like old wives' tales.
So, to prove the typical way is to introduce a fresh and prove . After that, the that was introduced, disappears.
They are not propositional equivalent, first, because (f x) is not a proposition since it has a free variable x. The truth value of (f x) depends on x.but it then says that they are not propositional equivalent
Now, thinking about this a bit more, I would say that they are not propositional equivalent because they can't be proved equivalent using propositional rules. One can prove from , but only by eliminating from the discourse and only if was fresh. This inference rule is from first-order, not propositional, logic.