Is the product of conditional posterior equal to the joint distribution?

Given conditional posterior probabilities of $x$ random variables, can we find the joint distribution by multiplying the conditional posteriors together ? An example would be distribution $$p(x_1,x_2,x_3) = p(x_1|x_2,x_3)p(x_2|x_1,x_3)p(x_3|x_1,x_2) , ?$$ I know that the chain rule of probability would result in $$p(x_1,x_2,x_3) = p(x_1|x_2,x_3)p(x_2|x_3)p(x_3)$$