Tautologies in First Order Logic So I am asked to put together a tautology that accurately reflects a substitution instance in First Order Logic for the the following sentence:  
“If I have two dollars, I can buy a soda, and if I have two dollars, then I can buy a candy bar, so if I have two dollars, I can buy a soda and a candy bar”
I have come up with the following: ((A → B) ∧ (A → C)) → (A → (B ∧ C))
Then I am asked to give an informal proof that demonstrates that it is a tautology. How do I do an informal proof?
 A: Distinguish "I can buy-a-soda-and-a-candy-bar" from "I can buy a soda and I can buy a candy bar".
With $A$ for "I have two dollars", $B$ for "I can buy a soda" and $C$ for , "I can buy a candy bar", 
$$((A \to B) \land (A \to C)) \to (A \to (B \land C))$$
says "Given if I have two dollars, I can buy a soda, and if I have two dollars, then I can buy a candy bar, then if I have two dollars, I can buy a soda and I can buy a candy bar”. Which is a tautology, and intuitively fine (if you have two dollars, you can buy the soda, and you can buy a candy bar -- nothing is said about your ability to buy both at the same time!).
But that wff doesn't transate the sentence you gave on the natural reading, as the natural reading of "I can buy a soda and a candy bar" (hooray, I can get both my treats!) is not rendered by $(B \land C)$.
A: 

“If I have two dollars, I can buy a soda, and if I have two dollars, then I can buy a candy bar, so if I have two dollars, I can buy a soda and a candy bar”

I have come up with the following: ((A → B) ∧ (A → C)) → (A → (B ∧ C))

Well, the argument itself is unsound, as you could not simultaneously spend the same two-dollars on different purchases, but anyway, yes, that is an accurate rendition of the statement you were told to translate.
So, to prove that the statement is a tautology we just note informally that : If it is true that both $(A\to B)$ and $(A\to C)$ hold simultaneously, then it would be so that if $A$ held then both $B$ and $C$ would hold.  
$\\[12ex]$

A sounder argument would be: "I may buy a soda if I have two dollars, or I may buy a candy bar if I have two dollars, so I may buy a soda or a candy bar if I have two dollars."
