Consider two logical statements for real numbers $x,y,z$
$\forall x (\forall y \exists z (y = xz) \implies (x \neq 0))$
$\forall x \forall y \exists z ((y = xz) \implies (x \neq 0))$
In some course notes on logic I am instructed that 1) is true and 2 is false
For 1) I believe there is two cases. Suppose $x=0$, then the statement $\forall y \exists z (y=xz)$ is false for $y=10$, and so we get False implies True, which is True. If $x \neq 0$ then we get True implies True, which is True.
For 2) I'm not sure why I can't simply use the same reasoning, or what the real difference is between 1 and 2)
Any insights appreciated.