For two continuous random variables X and Y, the joint probability density function is

fX,Y(x,y) = fY|X(y|x) fX(x) = fX|Y(x|y) fY(y)

where fY|X(y|x) and fX|Y(x|y) are the conditional distributions of Y given X=x and of X given Y=y respectively, and fX(x) and fY(y) are the marginal distributions of X and Y respectively.

If we pose the restrictions that both of the above marginal distributions are normal, AND the conditional distributions are ALSO normal.
does this fix the joint distribution fX,Y(x,y) HAS to be bivariate normal?

Can anyone show a proof whether the joint distribution HAS to be bivariate normal or not necessarily?