I think that your solution is ok. You may notice that the last line is in fact .

You can also use truth table to verify that your answer is correct.

For proposition which have the form of implication there is a shorter way:

Clearly if false only in one case: p=1, q=0.

Let us try to the same thing with the first expression

.

This implication is false only if and q=0. For a conjunction to be true, you need that both of the operands are true. This means that p=1, and is also true for p=1,q=0.