To prove the logical equivalence $\neg (p \land \neg \neg (p \lor q)) \equiv \neg p$, we can follow these steps:

1. Double Negation Law: $\neg \neg A \equiv A$ Apply the double negation law to $\neg \neg (p \lor q)$: $\neg \neg (p \lor q) \equiv (p \lor q)$

   Substituting this back into the expression, we get: $\neg (p \land (p \lor q))$

2. Distributive Law: $A \land (A \lor B) \equiv A$ Simplify $p \land (p \lor q)$: $p \land (p \lor q) \equiv p$

   So, the expression becomes: $\neg p$

Thus, we have shown that: $\neg (p \land \neg \neg (p \lor q)) \equiv \neg p$

Therefore, the logical equivalence $\neg (p \land \neg \neg (p \lor q)) \equiv \neg p$ is proven.

Would you like more details or have any questions?

Here are 8 related questions that might interest you:

1. How do you prove logical equivalences using the truth table method?
2. What are the main differences between propositional and predicate logic?
3. Can you explain how the implication $(p \implies q)$ is equivalent to $(\neg p \lor q)$?
4. How do logical equivalences help in simplifying Boolean expressions?
5. What is the importance of the associative law in logic?
6. Can you give an example of using De Morgan's laws in a logical proof?
7. How does the concept of logical equivalence apply to programming?
8. What role do logical operators play in digital circuit design?

Tip: When working with logical expressions, it's useful to become familiar with the most common laws and equivalences in logic. This familiarity can significantly speed up the process of simplifying and proving logical statements.