I want to prove that

A /\ (B V C) => (A /\ B) V (A /\ C)

using the Fitch system.

I have done the following:

1. A /\ (B V C)

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2. A /\ Elim : 1

3 B V C /\ Elim: 1

4. B

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5. A /\ B /\ Intro: 2, 4

6. C

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7. A /\ C /\ Intro: 2, 6

8. (A /\ B) V (A /\ C) V Intro : 3, 4-5, 6-7

Can I do step 8)? Or am I cheating here? How should I do this?

From what I've read on Logic and Proof, it gave me the impression that you must always conclude the same thing in all "paths" of the OR. I am not doing that, so that is why I'm a bit confused on how to approach this.

Thanks