Difficulty understands logic

Question

So, I have to prove the following sequence:

(p -> r) ^ (q -> r) |- p ^ q -> r 

I understand why this is correct, and I also understand the rules of natural deduction. I do not understand how I prove it. Here is an example of a model response:

 1. (p -> r) ^ (q -> r) |- p ^ q -> r premise 2. p ^ q assumption 3. p ^e 2 4. p -> r ^e 1 5. r ->e 4,3 6. p ^ q -> r ->i 2,5 (e = elimination / i = introduction). 

Can someone provide me with a link or explanation? I feel that I am missing a simple concept that makes it hard to understand ...?

For example, on line 4, why do you need p from line 3 to delete →, where, as on line 3, can you delete ^ q without using q?

I'm sure this is pretty straight forward, but for me it doesn't make sense ...?

Answer 1

In line 2, you have p ^ q , which means that both p and q are true. It follows that p true, because if both of them are true, then either of them is also true.

In line 4, r true only if p true. And on line 3, you have p true. Therefore, r is also true.

Answer 2

You can remove ^ q without using q, because p ^ q means that p AND q - p is true regardless of q.

You cannot delete p → without using p, because p → r means p IMPLIES. r - r is guaranteed to be true if p also.