I need some help with the follwing: Lets suppose that the sentence $\forall x: x\in I \rightarrow P(x)$ is false. Now consider the sentence $\forall x: x\in I \rightarrow (P(x) \ \& \ Q(x) ) \ \ \quad (1)$ for any property $Q(x)$, which is also obviously false. But now I can define a set I'=\left\{ x \in I| P(x) \right\}. So the sentence (1) should be equivalent to the sentence \forall x: x\in I' \rightarrow Q(x) But since sentence is now true, since there aren't any $x$ such that x \in I', because the first sentence was supposed to be false.
I'm sure the error is, that the two sentences are equivalent. But I can't pinpoint my error. Could someone tell me crystal-clear what I am doing wrong ?