Translating Predicate Logic to English 
*

*$E(x,y)$: $x$ can eat $y$

*$L(x,y)$: $x$ loves eating $y$

*$D$ is the domain of all dogs

*$S$ is the domain of all snakes


Predicate Logic to English:  
                             
  $\forall a \in S,\sim \ \bigg[   \exists c \in S,\ a\ \ne  c  \ \wedge E(a,c)\bigg] \iff \forall b \in D, \ L(a,b)$:  


All snakes, a, cannot eat any other snakes , if and only if, all snakes, a, loves eating all dogs. 

Any thoughts on if this is accurate and/or a way to condense this statment?
 A: Start with the part in the square braces:

There exists a snake $c$ such that $a$ is different from $c$ and $a$ can eat $c$.

Now negate it:

For all snakes $c$, either $a$ and $c$ are the same, or $a$ cannot eat $c$.

or more plainly,

$a$ cannot eat any snake that isn't itself

Now, the right-hand side:

For all dogs $b$, $a$ loves eating $b$.

The statement then asserts that for every snake $a$, either both or neither of the two preceding statements hold, i.e. 

For every snake $a$, $a$ cannot eat any other snake if and only if $a$ loves eating every dog.

Notice the big difference between my 'translation' and yours is that mine applies case by case: for each snake $a$ either both or neither are true, but results may differ across snakes.
A: 
Not all snakes is the same as some other snake, and that not all snakes can eat some other snake, if and only if, all snakes loves eating all dogs. 

Note: The scope for the first universal binding (on $a$) seems intended to be the entire biconditional. $~$ This should be parenthesised, because biconditionals have operational precedence over quantification.$$\def\iff{\leftrightarrow}\forall a {\in} S~\color{blue}{\big(}\lnot \big[\exists c {\in} S~ a\neq  c \land E(a,c)\big] \iff \forall b {\in} D~L(a,b)\color{blue}{\big)}$$
As such, say "All/Any snake" once, applying it to the entire sentence, rather than using it for each phrase.   Likewise for the negated existential.   When it is written once, say it once.
"Any snake, cannot eat any other snake if, and only if, it will love eating every dog."
