On Wikipedia on page relative to Conditional probability in section Statistical independence it is written following
It should also be noted that given the independent event pair [A B] and an event C, the pair is defined to be conditionally independent if the product holds true:[17]
$$P(AB\mid C)=P(A{\mid}C).P(B{\mid}C)$$
Is that equivalent to say that
If A and B are two mutually independent events, than
$$P(A\cap B\mid C)=P(A{\mid}C).P(B{\mid}C)$$
Is that true ?
How can I prove the last formula ? Wikipedia is citing her source, but cited book are not readable online !
PS: I have "corrected" Wikipedia sentence today. Perhaps that my correction is incorrect !
Last formula is not true in general. For example taking $$\Omega =\{ 1,2,3,4 \}, A = \{1, 2\}, B=\{ 2,4 \}, C=\{1,2,4\},$$ with classical probability give you pair of independent events $A, B$. But $$P(A\cap B| C) = \frac 1 3$$ and $$ P(A| C)P(B| C) = \frac 2 3 \cdot \frac 2 3 = \frac 4 9.$$
This shows that your definition of conditionally independent events actually introduces something different to regular independence. It could be thought of as independence in restricted space $C$.