How to Prove the Logical Implication from Church's Mathematical Logic?

[Hessinger]

This from Alonzo Church's Mathematical Logic, been stuck on it for a week =(.

Homework Statement

14.3 Present a Formal Proof: p $$\Rightarrow$$ (q $$\Rightarrow$$ r) $$\Rightarrow$$ ((p $$\Rightarrow$$ q) $$\Rightarrow$$ r)

Homework Equations

The Attempt at a Solution

A truth table has shown that the previous implication is a tautology therefore we should be able to prove it. The first half is easily obtained from modus ponens... p $$\Rightarrow$$ (q $$\Rightarrow$$ r) however I have not been able to get ((p $$\Rightarrow$$ q) $$\Rightarrow$$ r) any suggestions or guidance would be appreciated.

 

[Elucidus]

I assume you are asked to show $p \rightarrow (q \rightarrow r) \Rightarrow (p \rightarrow q) \rightarrow r$.

Proofs involving conclusions of the form "if A then B" are usually best proven by assuming the premises of the claim and A and then showing B is a consequence.

Basically:

Given $p \rightarrow (q \rightarrow r), (p \rightarrow q)$.

Show $r$.

As a hint, I'd suggest assuming p as a first step in the proof.

--Elucidus

P.S.: This method is valid due to the equivalence $(A \wedge B) \rightarrow C \equiv A \rightarrow (B \rightarrow C)$