What is the difference between
$\forall x,\exists y, ((x,y) \in A \implies x \in B)$ and $\forall x,((\exists y, (x,y) \in A) \implies x \in B)$ ?

I understand that the second means that $pr_1 A \subseteq B$, but what about the first sentence ?


closed as off-topic by Yemon Choi, Andreas Blass, Emil Jeřábek, Alexey Ustinov, Matt F. Sep 11 at 14:27

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  • 2
    $\begingroup$ The first sentence is equivalent to $\forall x\,((\forall y\,(x,y)\in A)\implies x\in B)$. $\endgroup$ – Andreas Blass Sep 11 at 14:18

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