Prove the logical equivalence ￢ 𝑝 → ￢ 𝑞 ≡ 𝑞 → 𝑝 using equivalencelaws.

Sure, here is the step-by-step proof of the logical equivalence ¬𝑝 → ¬𝑞 ≡ 𝑞 → 𝑝 using equivalence laws:

1. Start with the left side of the equivalence: ¬𝑝 → ¬𝑞
2. Apply the implication law, which states that 𝑝 → 𝑞 is equivalent to ¬𝑝 ∨ 𝑞. This gives us ¬¬𝑝 ∨ ¬𝑞.
3. Apply the double negation law, which states that ¬¬𝑝 is equivalent to 𝑝. This gives us 𝑝 ∨ ¬𝑞.
4. Apply the commutative law, which allows us to swap the order of the disjunction. This gives us ¬𝑞 ∨ 𝑝.
5. Apply the implication law again, this time in reverse, to get 𝑞 → 𝑝.
6. Therefore, ¬𝑝 → ¬𝑞 is logically equivalent to 𝑞 → 𝑝.