Is it legal to change
$\exists$x A(x) $\implies$ $\exists$y B(y)
to:
$\exists$x $\exists$y (A(x) $\implies$ B(y))
If i'm thinking correctly, A(x) is free from $\exists$y and should be allowed to move. However i am not able to prove it. So far i have:
$\neg$$\exists$x A(x) $\vee$ $\exists$y B(y)
$\forall$x $\neg$A(x) $\vee$ $\exists$y B(y)
$\forall$x $\neg$A(x) $\vee$ $\neg$$\forall$y $\neg$B(y)
$\forall$x ($\neg$A(x) $\vee$ $\neg$$\forall$y $\neg$B(y))
$\forall$x $\neg$ $\forall$y($\neg$A(x) $\vee$ $\neg$B(y))
$\neg$$\exists$x $\forall$y($\neg$A(x) $\vee$ $\neg$B(y))
$\neg$$\exists$x $\forall$y(A(x) $\wedge$ B(y))
$\neg$$\exists$x $\neg$$\exists$y $\neg$(A(x) $\wedge$ B(y))
$\neg$$\exists$x $\neg$$\exists$y ($\neg$A(x) $\vee$ $\neg$B(y))
$\neg$$\exists$x $\neg$$\exists$y (A(x) $\implies$ $\neg$ B(y))
I'm likely to have mistakes here, have not touched this in a while.