I solved this exercice that I need to deliver today but I'm not really sure if the way I did it is correct.
I started by proving (δ→φ)∧(φ→δ)⊢(δ∨φ)→(δ∧φ) and then proving (δ∨φ)→(δ∧φ)⊢(δ→φ)∧(φ→δ) as you can see on the image:
It would mean a lot if you can tell me if the way I did it is correct or not, thanks.
For the first proof, on line 9 you refer to line 5, but that is not allowed, since line 5 is part of a subproof is closed off to line 9.
What you need to do is to do a proof by cases on the disjunction you have on line 2 ... Note that indeed you never used line 2 in your proof, which should strike you as pretty weird in the first place, since that is the antecedent of the conclusion: surely you would need to use that at some point!
So: on line 3, assume $\phi$, and using the left conditional from line 1, get $\psi$, and then conjunct the two to get $\phi \land \psi$. Then do a second subproof assuming $\psi$ and eventually get $\phi \land \psi$ as well, and then by v Elim you get $\phi \land \psi$ in the outside subproof.
The second proof is correct!