Translating $\forall$ and $\exists$ in a statement The following statement has two versions – one where $d$ is quantified by $\forall$ and the second where it's quantified by $\exists$. The task here is to find a counterexample  where the statements below are false. The domain is all integers.


*

*$\forall a \forall b \forall c \forall d(a^d + b^d = c^d$)

*$\forall a \forall b \forall c \exists d(a^d + b^d = c^d$)


The first statement is false for at least some values of the variables. When $a=1, b=2, c=3, d=4$, the statement does not hold for all variables $a$, $b$, $c$, and $d$. For instance: 
$$
1^4 + 2^4 \ne 3^4
$$
The second statement is false when $a = 1, b = 2, c = 10$ because there doesn't exist a $d$ where the statement would be true. For instance, if $d$ was $5$:
$$
1^5 + 2^5 \ne 10^5
$$
Am I interpreting this correctly and do my counterxamples make sense?

Turning a comment into an answer:
For the first statement, you can remove the for all variables a, b, c, and d part, as the counterexample of $[a=1,b=2,c=3,d=4]$ already states this fact.
For the second statement, you need to prove that your counterexample of $[a=1,b=2,c=10]$ is true for ALL values of $d$.
I'm not sure whether or not it is indeed true for all values of $d$, but it is certainly not so trivial.
Alternatively, you can use the counterexample of $[a=0,b=0,c=1]$ which is a lot more trivial.
