I need help with the following proof using Rules of Inference:
∃x(A(x)∧¬B(x))
∀x(A(x)→C(x))
Therefore: ∃x(C(x) ∧ ¬B(x))
We can use Rules of Inference Such as
Modus Ponens
Modus Tollens
Hypothetical Syllogism
Addition
Simplification
Conjunction
Disjunctive Syllogism
It seems to me that the problem statement is clear. However, I would also recommend writing an informal proof in English first and then translate it into inference rules.
A thing to remember is that each connective has rules for using it (when a formula with this connective is given as an assumption) and rules for deriving it. So here you need to use one existential formula and derive another. One must understand restrictions on each rule. For example, if is is given that "There exists a movie that is an action movie and is not 'Twilight'", one can't conclude, "Without loss of generality, we can assume that the movie in question is 'Ninja Assassin'". However, when we are asked to prove that "There exists a movie that is an action movie and is not 'Twilight'", then we can say, "Consider 'Ninja Assassin', for example".
Of course, you need not only propositional rules but rules dealing with quantifiers as well, such as Universal Introduction/Elimination and Existential Introduction/Elimination.
These are some generic thoughts. If you have some particular difficulty, please describe it.