Existential generalization of a statement function.

Question

1. ( x )( Ax —> Bx)
2. Ay —> By
3. ( ∃x )(Ax —> Bx)

Line 1 is the premise, line 2 is obtained through universal instantiation of line 1 and line 3 is obtained through existential generalization of line 2.

Is the inference at line 3 valid?

I read in my logic book that if a statement function is instantiated with respect to a constant or a variable, then you can existentially generalize the statement. So, according to the mechanical rule given by the book, I am justified to make the inference at line 3. In other words, since line 2 is instantiated with respect to the variable 'y', then I am justified to make the existential generalization at line 3.

My intuition tells me that it's wrong to make the inference at line 3, because such inference means that I am asserting the existence of something which is not asserted to exist by our premise. Our premise only asserts that if an element belongs to class A, then this element must also belong to class B. Thus, the premise neither asserts that class A has elements nor it asserts that class B has elements. Now, if the inference at line 3 was valid, then I will be asserting the existence of at least one element that belongs to both classes. But such inference can't follow 100% perecent from the premise, thus I see the inference to be invalid.

Answer 1

When the empty domain is permitted, it must often be treated as a special case. Most people, however, simply exclude the empty domain by definition.
That is, most commonly, in first order logic, we always have at least one element in the universe.

Also, your statement

if the inference at line 3 was valid, then I will be asserting the existence of at least one element that belongs to both classes

is incorrect.

"At least one element in both classes" would be $\exists x (Ax \land Bx)$.
$\exists x (Ax \implies Bx)$ means something different, use the truth table semantics of $\implies$ to understand the difference.