The Fitch-style natural deduction proof checker and editor I am using for this answer is associated with the book forall x: Calgary Remix.
Here is the question:
1......1. p.(q V r)
1......2. q V r ... .....................(1) CE
1......3. p ...............................(1) CE
I need two "p" for the conclusion, how can I introduce another "p" and
keep it in the conclusion?
The solution below is similar to what Bram28 provided in lines 2-11:
To get past line 3, we need to eliminate the disjunction, the ∨ symbol. This is an "or" statement. Either Q is true or R is true. So to eliminate the "or" we need to consider two cases. I drew thin blue boxes around the two cases, one for Q and one for R.
Regarding the question about needing two "p" for the conclusion, the extra "p" is added in lines 6 for the Q case and in line 9 for the R case.
Note how this was done in the Q case.
In line 4 I started a sub-proof by assuming Q. I need no justification for that assumption.
In line 5 I used the fact that in line 2 I already have P and in line 4 I have Q as an assumption. Since I have both of them I can introduce a conjunction, that is, an "and" statement. Now I have P ∧ Q, part of the conclusion I want.
After that I can introduce a disjunction, that is, an "or" statement to the P ∧ Q. What will I add? I can add anything I want. I already know this statement is true because one of the cases, P ∧ Q, is true. So I introduce the ∨ with precisely what I need to get the result I want: P ∧ R.
I've taken care of the two cases by constructing a sub-proof for each one and in each case I reached the desired conclusion. The proof will be complete once I claim that. In line 10 I state the conclusion from both sub-proofs. The justification of this is an elimination of the disjunction I started with in line 3 using sub-proofs in lines 4-6 and 7-9.
The proof checker confirms the solution.
We can go in the other direction as well. Bram28 does this in lines 12-23 of that proof. The last line of that proof introduces a biconditional by referencing as justification for the introduction the two sub-proofs on lines 2-11 and 12-23.