I am in a situation where I have a space Y obtained from a space X by gluing on infinitely many trivial cells. That is, I have a collection of maps
$f_\alpha: A_\alpha \to X$
each of which is a cofibration. I also have weak equivalences
$g_\alpha: A_\alpha \to B_\alpha$.
And I know that $Y$ is obtained as the pushout of
$\coprod_\alpha B_\alpha \leftarrow \coprod_\alpha A_\alpha \to X$.
Now, I know that a pushout of a weak equivalence along a cofibration is again a weak equivalence. But here, while each map $f_\alpha$ is a cofibration, the coproduct $\coprod_\alpha f_\alpha$ is not a cofibration (the images overlap, for instance).
It seems like the induced map $X \to Y$ should still be a weak equivalence. How do I prove this?