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Expression - (A OR B OR C OR D) AND (!B AND !D)

I know that with distributive property, it holds that (a OR b) AND (c OR d) = (a AND c) OR (a AND d) OR (b AND c) or (b AND d) but I'm not sure how it will work if the second group has an AND

Steps in the answer would help.

Kaushik Evani
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    While boolean logic is often used in programs and programming, this seems more to be about the mathematical properties of boolean algebra. As such it should probably be posted on [the Math SE site](https://math.stackexchange.com/tour). – Some programmer dude Nov 06 '19 at 09:21

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Something like this perhaps?

  (A OR B OR C OR D) AND (!B AND !D)
= (A OR B OR C OR D) AND !(B OR D)
= ((A OR C) OR (B OR D)) AND !(B OR D)
= ((A OR C) AND !(B OR D)) OR ((B OR D) AND !(B OR D))
= (A OR C) AND !(B OR D) OR false
= (A OR C) AND !(B OR D)

That seems like it's going to be minimal since each variable appears once and there are no apparent contradictions or tautologies.

Patrick87
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