Using law of logic, simplify the statement form $A \lor [\neg(\neg A)\land B]$

Question

Simplifying the given expression led me to $A\lor\neg B$. Here is what I did:

$A \lor [\neg(\neg A)\land B]$ -------> Given

$A \lor [\neg(\neg A)\lor \neg B]$ ------> De Morgan's Law

$A \lor (A\lor \neg B)$ -----------> Law of double negation

$(A \lor A)\lor \neg B$ -----------> Associative Law

$A \lor \neg B$-----------------> Idempotent Law

Now, I want to know if my application of De Morgan's Law (second step) is correct. Thanks!

Answer 1

Everything is fine, but you may simplify even further : by definition of $\to$, the last line is equivalent to $$B \to A$$