Having trouble with this proof

Question

I have to prove the following: (((p ∨ q) ≡ (r ∨ s) ≡ (p ∨ q ∨ r ∨ s)) ∧ t ∧ u) → (r ∨ s). The proof is supposed to be relatively simple according to the professor but I have no clue where to start. Thanks.

Answer 1

Let $A$ be $p\lor q$, $B$ be $r\lor s$, and $C$ be $t\land u$,. In order to prove $$(((A\leftrightarrow B)\leftrightarrow(A\lor B))\land C)\rightarrow B,$$ it is sufficient to prove $$(P\leftrightarrow Q)\rightarrow B,$$ where $P$ is $A\leftrightarrow B$ and $Q$ is $A\lor B$.

Since $\neg Q\rightarrow P$, we have $(P\leftrightarrow Q)\rightarrow(P\land Q)$.

Moreover, $(P\land Q)\rightarrow B$, which ends the proof.