I interpreted this as a case of the extension of De Morgan's Law to quantifiers. https://en.wikipedia.org/wiki/De_Morgan%27s_laws#Extensions
I know that similar questions have been asked before about a proof for De Morgan's Law for quantifiers, but this one is a bit different.
I couldn't find a proof for the quantifier-extension of De Morgan's Law that worked for a set of any cardinality so I have attempted to prove it using induction and I want to know if it is sound.
In the proof, I have denoted De Morgan's Law as DE and used it as a rule of inference. I also denoted the act of liking Reggae music as the function R(x)
I had to use an image to avoid text being displayed incorrectly. Also let me know if you know of a more elegant proof.
Explanation of step 18 I proved that De Morgan's Law could be used on an entire set with a cardinality of 3 (in step 9). Then I proved that if DE is done on an entire set with a cardinality of $n – 1$ (step 11), then DE could be done on a set with a cardinality of $n$. This is the requirements needed for induction, thus step 18 is valid. Also take note of step 13.
Let me know if the proof is sound.
Actually, I think that the clumsy statement "Everyone does not like Reggae music" in a normal English context is the same as the clearer "Not everyone likes Reggae music", which of course is not equivalent to "No one likes Reggae music". I admit there is some ambiguity about such usage, which is why it should be avoided.