I am trying to understand how the internal language of a topos works, but there are few examples in the various sources I read and in particular I found no examples related directly to specific objects of the topos, like the terminal object $1$ or the subobject classifier $\Omega$. For example, are the following judgements valid in every topos?
$$ x : 1, y : 1 \vdash x = y\tag{1} $$
$$ \vdash \exists w : \Omega, w = w\tag{2}$$
$$ x : 0 \vdash \bot \tag{3}$$