using the elimination rule in natural deduction Prove that
$$(A ∧ B) \to C ⊢ A \to (B \to C)$$
Am I using the conjuction elimination rule correctly? Or am I assuming too much?

  
*
  
*$(A ∧ B) \to C$  (Given)
  
*$A \to C , B -> C$ (∧E 1)
  
*$A \to( B \to C)$ ($\to$I 2) (QED)
  

 A: (1) The Conjunction Elimination Rule
In a standard natural deduction system, the Conjunction Elimination Rule states that

$$\frac{\varphi \wedge \psi}{\varphi} \frac{\varphi \wedge \psi}{\psi} \tag{$\wedge$E}$$

Which means we are allowed to apply it in sentences with a conjunction as its logical structure.
But you are not allowed to apply this rule in the above case. The sentence you have above is not a conjunction, but an implication. You are not using conjunction elimination correctly.

(2) Your Answer
In order to prove this statement, it suffices to see the logical form of your goal:
$$A \to (B \to C)$$
this suggests that in order to prove  this statement, we first assume $A$ as hypothesis, and eventually assume $B$ too, in order to obtain C:


*
  
*$(A ∧ B) \to C$, P


    
*$A$, H
    
    

      
*$B$, H
      
*$A \wedge B$, 2, 3 $\wedge$I
      
*$C$, 1, 4 $\to$E (Modus Ponens)
      
*$B \to C$, 3-5 $\to$I
      
*$A \to (B \to C)$, 2-6 $\to$I
      
      
      
    


Keep working!
