Paradoxical formula on conditional expectation.

Question

An accepted formula that is applied successfully in many conditional probability problems is: $$E[f(X,Y)\mid Y=t]=E[f(X,t)\mid Y=t]$$ This formula says that the conditional expectations $E[f(X,Y)\mid Y]$ and $E[f(X,t)\mid Y]$ agree at $\quad$ "$Y=t$". But what sense does that equality make when we know that conditional expectation is a global concept, not a pointwise one? The formula would make sense if we could say, if at all, that the equality holds for almost all $Y=y$, not just $Y=t$, but this is not true. It sounds like an absurd equality, and yet it works. How do we then have to interpret that equality? When does it make sense to apply it? Why does it work?