conditional of a conditional

Question

I have random variables $x,y,z$. Let $(x\mid y)$ denote a random variable whose distribution is the conditional distribution of $x$ given $y$. (Is there a standard notation for this object?) I would like to prove something like: $$ (x\mid y) \mid (z\mid y) = (x\mid y,z) $$ This seems intuitively true, but I'm having trouble proving it formally. I'm also not entirely sure that the notation I introduced even makes sense. I'm familiar with the law of total expectation, and this is similar, but I want a statement about the distributions, not just their conditional expectations.