Construction of a derivation when proving equivalence of a logical formula. Given $$A \wedge B \to C \equiv A \to B \to C$$
We want to show that $$\{ A \to B \to C\} \vdash A \wedge B \to C$$
by constructing a derivation using the natural deduction system $\mathcal{N}_{PL}$.
My question: 
How do i write down such a derivation?
So far I just came up with a "prove" by transformation:
\begin{align}
&A \to (B \to C)\\
&A \to (\neg B \vee C)\\
&\neg A \vee \neg B \vee C\\
&(\neg A \vee \neg B) \vee C\\
&\neg (\neg A \vee \neg B) \to C\\
&A \wedge B \to C
\end{align}
 A: First, your expression $A\rightarrow B\rightarrow C$ is not a well formed formula, you need to bracket it somehow and it makes a difference - the material conditional isn't associative. 
Now, when they say to prove it using natural deduction, they mean to prove it using only a specific set of rules, namely the rules of natural deduction. 
Read these 
https://www.cs.cornell.edu/courses/cs3110/2013sp/lectures/lec15-logic-contd/lec15.html
http://logicmanual.philosophy.ox.ac.uk/jsslides/ll6.pdf
A: 
So, do I get it right that I can just assume things, as follows:

*

*Assume $A$

*Then $B \to C$ (1.)

*Assume $B$

*Then $C$ (2., 3., Modus ponens)

*Then $A \wedge B$ ($\wedge$-introduction, 1., 3.)

*Then $A \wedge B \to C$ (2., 5.)


No, you do not "just assume things".   You start with a premise, them make and discharge assumptions as required to derive the proof.   Since to prove $(A\wedge B)\to C$, from $A\to(B\to C)$, the assumption you need to discharge will be $(A\wedge B)$, therefore that is the assumption you shall need to make.

*

*$A\to (B\to C)$ a premise

*$\quad A\wedge B$ by assumption

*$\quad A$ by conjunction elimination ($\wedge$E$_1$, 2)

*$\quad B\to C$ by conditional elimination ($\to$E, 1,3) aka modus ponens

*$\quad B$ by conjunction elumination  ($\wedge$E$_2$, 2)

*$\quad C$ by conditional elimination ($\to$E, 4,5) aka modus ponens

*$(A\wedge B)\to C$ by conditional introduction ($\to$I, 2,6)

Where as when starting with the premise $(A\wedge B)\to C$, there you need to discharge the two assumptions $B$, $A$ to prove $A\to(B\to C)$, so

*

*$(A\wedge B)\to C$ a premise

*$\quad A$ by assumption

*$\qquad B$ by assumption

$~\vdots$ $~~~\qquad \vdots$
? $~~~\qquad C$ by reasons
? $~~~\quad B\to C$ by conditional introduction ($\to$I, 3,?)
? $~~~A\to (B\to C$ by conditional introduction ($\to$I, 2,?)
