"Would there be a chance of a different but intelligent life evolving in our place"?
Of course there would be "a chance", even though we don't know what that chance would be. If humans evolved to get intelligent life, then evolving an intelligent life is not impossible, which means that it could happen again.
(even though I wager that the probability is little, because on millions of years of evolutionary life, with so many different species and so many different environments, only humans evolved to this point)
In fact, if humans went away, this evolution would be, IMHO, more probable, since human beings consume too much resources to allow another intelligent species to arise. Human beings are so much ahead on that department, with their technology and knowledge, that there is just no evolutionary benefit in competing with them for it... just in adjusting to the environmental changes that humans produce.
Now, we need to focus on another thing. Your question was not only about "intelligent life", but also "different". On this regard, maybe "different" would be harder to achieve, since other primates, like chimps, are the second more likely contesters to develop intelligent life (in fact, many people here may argue that they already developed intelligent life, even though I don't think they are comparable).
So it would be likely that we would see the emergence of another intelligent primate species.
But then again, you said that in your world, the environment has become too much deadly for us. You don't say how, or hor much, so I may only guess. But this would be probably also deadly for primates, which could go extinct. So we could go on to the next species... are the oceans compromised? If not, why not consider dolphins? They are already pretty intelligent.
Also, if the environment is really screwed up, maybe insects would have the evolutionary upper hand. On that regard, maybe ants or bees could be a good choice, since their gregarious nature would be more likely to develop a kind of language, which is fundamental for intelligence.
Finally, you could ponder an alternative... if Earth has become too much deadly for humans, maybe it was because of the humans own doing. So, maybe those humans could have genetically altered a non-human species before going away... which, left to its own devices, could have evolved intelligence more easily. Or (why not?), go all WALL-E and put the abandoned machines and computers evolving their own intelligence?