1. **State the problem:** We are given that $p \leftrightarrow q$ is true, and we need to show that $p \lor \neg q$ is equivalent to $q \lor \neg p$. 2. **Recall the meaning of $p \leftrightarrow q$:** This means $p$ and $q$ have the same truth value. Formally, $p \leftrightarrow q$ is true if and only if $(p \land q) \lor (\neg p \land \neg q)$ is true. 3. **Analyze $p \lor \neg q$ under the condition $p \leftrightarrow q$:** - Since $p$ and $q$ have the same truth value, if $p$ is true, then $q$ is true. - If $p$ is true, then $p \lor \neg q$ is true because $p$ is true. - If $p$ is false, then $q$ is false (because $p \leftrightarrow q$), so $\neg q$ is true. - Therefore, if $p$ is false, $p \lor \neg q$ is true because $\neg q$ is true. 4. **Analyze $q \lor \neg p$ under the same condition:** - If $q$ is true, then $q \lor \neg p$ is true. - If $q$ is false, then $p$ is false (because $p \leftrightarrow q$), so $\neg p$ is true. - Therefore, if $q$ is false, $q \lor \neg p$ is true. 5. **Conclusion:** Both $p \lor \neg q$ and $q \lor \neg p$ are true in exactly the same cases when $p \leftrightarrow q$ is true. Hence, $p \lor \neg q$ is logically equivalent to $q \lor \neg p$ under the condition $p \leftrightarrow q$. **Final answer:** $$p \lor \neg q \equiv q \lor \neg p$$