As you said, I tried to solve it explicitly and exactly.
The book doesn't explain about definition derivations in predicate calculi; It just adds two rules to the propositional rules of natural deduction. But if i guess right about the
atomic formulas that they are the BASIC step of the recursive definition of derivations, the first part of my proof seems to be true:
1-
Basics - Since (as we assumed above) the atomic formulas are themselves derivations too (with single nodes), then for any atomic formula phi we have a derivation (D ... phi) with (the only one) hypothesis phi itself. Now the desired property would obviously hold, because since (phi
|- phi) implies that (phi[x/c]
|- phi[x/c]).
2- For
propositional connectives, i believe that they behave similarly about the problem. So i just solved the (^I) case. [Attached pic 1]
3- For
universal elimination rule, i solved it without using the substitution theorem you mentioned. [Attached pic 2]
4- For
universal introduction rule, i couldn't use the trick (red starred around the green line!) in the previous solution, because here we have the problem of cancelling that hypotheses in Gama that contain x as a free variable. [Therefore not all sets that contain all hypotheses of the the final derivation (using univ introduction rule), would contain hypotheses of the first derivation (used as the inductive hypothesis)].
Sorry if bad handwriting! Ask on any ambiguity...
I'll wait to see the other solutions.