In the book
Discrete Mathematics and Its Applications by Kenneth H. Rosen, page 77
appears this example.
Example 13: Show that the premises “A student in this class has not read the book,” and “Everyone in this class passed the first exam” imply the conclusion “Someone who passed the first exam has not read the book.” Solution: Let C(x) be “x is in this class,” B(x) be “x has read the book,” and P (x) be “x passed the first exam.” The premises are ∃x(C(x) ∧ ¬B(x)) and ∀x(C(x) → P (x)). The conclusion is ∃x(P (x) ∧ ¬B(x)). These steps can be used to establish the conclusion from the premises.
There, the application of Existential Instantiation uses variable a and the usage of Universal Instantiation uses that same variable.
From (1), I can conclude $C(a) \wedge \neg B(a)$ is true for some a and from (4) I can conclude $C(a) \implies P(a)$ is true for any a ?
Are both variables conceptually different? i.e how can I know the second a is talking about the same object as the first a.
Any clarifications are very welcomed.