Proving a variable true through rules of inference

Question

Question: Use rules of inference to show that if $(p → q) ∧ (q → p),\; t ∨ q,\; t ∨ p,\; (p ∧ q) → t$, then $t$ is true.

Work So Far:

$$\text{1. }(p \implies q) \land (q \implies p)\text{ | Premise}$$ $$\text{2. }t \lor q\text{ | Premise}$$ $$\text{3. }t \lor p\text{ | Premise}$$ $$\text{4. }(p \land q) \implies t\text{ | Premise}$$ $$\text{5. }p \iff q\text{ | Biconditional Equivalency, Line 1}$$ $$\text{6. }t \lor (p \land q)\text{ | Distributive, Lines 2 and 3}$$

I'm basically stuck here on this problem. While it seems like the conclusions made on lines 5 and 6 could be useful, it doesn't feel like it really gets me anywhere in proving $t$ is true. I believe that I need to be able to pull the fact that a variable is true or false out of one or more of these premises and then use that in the other premises to eventually prove $t$ true, but I can't figure out how to pull any variable as true or false out of the four premises.

Answer 1

I am going to provide a very easy though long winded way of going about this. You start off by assuming that four statements are true. Denote these statements as follows:

• $\alpha : (p\to q)\land (q\to p)$
• $\beta : t\lor q$
• $\eta : t\lor p$
• $\gamma : (p\land q)\to t$

Now break the problem down into four cases based on the truth values of $p$ and $q$.

• Case 1: Suppose $p$ and $q$ are both true. Then $t$ must be true for $\gamma$ to be true.
• Case 2: Suppose $p$ and $q$ are both false. Then $t$ must be true for $\beta$ and $\eta$ to be true.
• Case 3: Suppose $p$ is true and $q$ is false. Then $t$ must be true for $\beta$ to be true.
• Case 4: Suppose $p$ is false and $q$ is true. Then $t$ must be true for $\eta$ to be true.

As seen in the four cases above, for all of $\alpha,\beta,\eta,\gamma$ to be true, we must have that $t$ is true in each case.