Rules of Inference on lines with multiple quantifiers

Question

Is one able to use rules on inference on lines containing multiple quantifiers, for example:

1. $(\exists x)Cx \to (\exists x)Ax$

2. $(\exists x)Ax \to \lnot(\exists x)Bx$

3. $(\exists x)Cx \to\lnot (\exists x)Bx$ H.S. 1,2

or would the quantifiers need be removed to make the inference?

Answer 1

What you do is fine!

Hypothetical Syllogism is defined as:

$$\varphi \rightarrow \psi, \psi \rightarrow \chi \vdash \varphi \rightarrow \chi$$

where $\varphi$, $\psi$, and $\chi$ are any logic formulas. Your formulas just happen to contain quantifiers.