Overview
Here is a summary of the rules of inference for conjunction, conditional and biconditional you will use below:
∧E You may write φ or ψ provided you have φ ∧ ψ in a prior line.
∧I You may write φ ∧ ψ provided you have both φ and ψ in two prior lines.
→E You may write ψ provided you have φ → ψ and φ in two prior lines.
→I You may write φ → ψ provided you have a subproof of ψ from φ.
↔︎E You may write ψ provided you have φ ↔︎ ψ and φ in two prior lines.
You may write φ provided you have φ ↔︎ ψ and ψ in two prior lines.
↔︎I You may write φ ↔︎ ψ provided you have one subproof from ψ to φ and one from φ to ψ.
Problems
1,) P, Q ∧ R ⊢ Q ∧ P
2.) P, Q ∧ R ⊢ R ∧ P
3.) P ∧ (Q ∧ R_1) , (P ∧ Q) ∧ R ⊢ Q ∧ R
4.) P ∧ (Q ∧ R) ⊢ (R ∧ Q) ∧ P
5.)P, P → (Q ∧ R) ⊢ R
6.) P → Q , P → R , P ⊢ Q ∧ R
7.) P → (Q → R) ⊢ Q → (P → R)
8.) P ∧ (Q → (R_1 ∧ R_2)) ⊢ Q → (P → R_1)
9.) P ↔ R ⊢ (P ∧ Q) ↔ (Q ∧ R)
10.) ⊤ ⊢ (P ∧ Q) ↔ (Q ∧ P)