Basically the same issue as #1825 but more general.

from typing import Union


def f(a: Union[str, int]):
    if a == 'a':
        reveal_type(a)  # expect str
        str.lower(a)

test.py:6: error: Revealed type is 'Union[builtins.str, builtins.int]'
test.py:7: error: Argument 1 to "lower" of "str" has incompatible type "Union[str, int]"; expected "str"

This works if the conditional is extended with isinstance(a, str).

As noted in the referenced issue, attention must be paid to __eq__ being overrideable, meaning it cannot safely be inferred that the type of X in X == Type() is Type (or a subclass thereof) without looking into __eq__ overrides in both classes. I do not know how widely this is used for any types or for str, but I can say that I have never seen this myself. It seems that some types are more likely to compare equal to custom classes than others (e.g. None vs list), however.