Question regarding using the natural deduction system

Question

I have the following:

Premise: ((V → ¬W) ∧ (X → Y))
Premise: (¬W → Z)
Premise: (V ∧ X)
         |- (Z ∧Y)

The part I want to know is how do I go about separating the second premise ¬W so that it becomes ¬Z or (W → Z)

I have heard of doing double negation (¬¬W → ¬Z) but still unsure how to proceed from here.

Answer 1

As said in the comments, you don't need to split the second premise. Here you can use the conjunction elimination ($X\land Y \vdash X$, $X\land Y \vdash Y$), conjunction introduction ($X,Y\vdash X\land Y$) and modus ponens ($X,X\rightarrow Y\vdash Y$).

If I number you premises :

1. $(V\rightarrow \neg W)\land (X\rightarrow Y)$
2. $\neg W\rightarrow Z$
3. $V\land X$

Now you can apply conjunction elimination on 1. and 3. which gives you

4. $V\rightarrow \neg W$

5. $X\rightarrow Y$

6. $V$

7. $X$

Here you can see that if you use modus ponens on 6., 4. you get $\neg W$. This $\neg W$ is on the left of premise 2. So again using modus ponens, you get $Z$.

To get $Y$, you must use modus ponens one last time with 7. and 5.

Conjunction introdution gives you $Z\land Y$.

Here is what I get in Fitch notation :