if $(\forall\ x \in \mathbb{R})(x=x) \rightarrow\ (\forall\ x\in \mathbb{R})(x/x = 1) $
But this is not true because $x$ cannot equal zero for the second statement. Why?
I understand that we cannot divide by zero, but aren't these statements logically equivalent?
They are not equivalent.
The truth values for $x=x$ and $\frac xx =1$ are dependent upon whether $x = 0$ or not and the truth tables are :
$\begin{array}\ x=0 &|&x=x&|&\frac xx =1\\T&|&T&|&F\\F&|&T&|&T \end{array}$
So that simply are not equivalent. They just aren't.
And $(\forall x\in \mathbb R)x=x$ is simply true. and $(\forall x\in \mathbb R)\frac xx=1$ is simply false.
....
However the statements $(\forall x\in \mathbb R\setminus\{0\})x=x$ and $(\forall x \in \mathbb R\setminus\{0\})\frac xx=1$ are equivalent.
That truth table has simply one line:
$\begin{array}\ x\in \mathbb R\setminus\{0\}&|&x=x&|&\frac xx = 1\\T&|&T&|&T\end{array}$
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Also the statements $(\forall x\in\mathbb R)x=x$ and $(\forall x\in \mathbb R)\frac xx=1$ OR $x=0$ are equivalent (and both true).
And the statements $(\forall x\in \mathbb R)\frac xx=1$ and $(\forall x\in \mathbb R)[x=x$ AND $x\ne 0]$ are equivalent (but both false).