How can I prove that this propositional logic statement a tautology is?

**Hypothetical syllogism:**

Hi everyone,

I want to prove that the above statement is a tautology, but I don't want to use logic tables for A, B and C. I have tried using the rules of replacement, but I can't make it work. The proposition is called the "Hypothetical Syllogism." Which you can 'read' from the statement pretty easily, ie. if (a implies b) and (b implies c) then *this* implies (a implies c). You might know it from essay writing? Anyway, doesn't this mean the proof should give the following?

If I do use a truth table, I get 'true' for every value of A, B or C, which means that the statement is a tautology, but this has nothing to do with my idea for the proof. Can there be an equivalent proposition that only gives true? Is there a way of simplifying the proposition or do you have to use a table in this case?

Sorry, everyone, this was my mistake, I was trying to create a simple equivalent proposition, when I only had to show that the proposition was a tautology. My apologies.