**High school conditional Probability, Please help**

As a part of a bit complicated high school conditional probability question, I devised the following argument to get my answer to agree with the given answer. Please tell whether this argument is valid, by providing links for any theorems or so.

Below,

Three genes M, N, O that are responsible for the color of eyes occur randomly among adults and one person can have only one of these genes. Among children, probabilities of having Brown or Black eyes given that Parents are combination of MM, MN, MO...etc are given separately.

P(Ci) = Probabilities of parents having random genes M,N,O joining to produce a baby.

i.e, C1=MM C2=MN C3= MO C4= NM C5=NN C6=NO.... C9=OO

P(A) = Probability of Both parents having Black eyes

P(B) = Probability of child having brown eyes

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P(A∩B)

= ⅀ P([A∩B] | Ci) P(Ci)

= ⅀ P(A | Ci) P(B | Ci) P(Ci)

= ⅀ P(A | Ci) (P(B∩Ci)

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Note, in the second step, I assumed

P([A∩B] | Ci) = P(A | Ci) P(B | Ci)

because, A and B are both events that depend on C only. So, I take it that when given C has occurred (when the sample space is restricted to C only), Events A|C and B|C can be considered independent of each other.

Is this argument correct? Please support your answer with links or references to any theorems.