Let $k=\mathbb{F}_2$ be the usual galois field with two elements  
integers, we consider 
<ol>
<li> the alphabet $X=\{x,y\}$
<li> the set of noncommutative series $k<<X>>>=k^{X^*}$ 
(i.e. all functions $X^*\rightarrow k$ with the convolution product)
<li> the augmentation character $k<<X>>>\rightarrow k$ and 
its kernel $\frak{M}$ (series without constant term)
<li> the free group with two generators $\Gamma=\Gamma(x,y)$
<li> the group morphism $\mu : \Gamma(x,y)\rightarrow 1+\frak{M}$
$$
\mu(x)=1+x\ ;\ \mu(y)=1+y
$$
which is known to be into (Magnus transformation)
<li> the quotients $\mathcal{A}_n=k<<X>>>/\frak{M}^n$ (which are finite) 
and the sujective quotient morphisms $q_n : k<<X>>>\rightarrow \mathcal{A}_n$
<li> the groups $\Gamma_n$, images of $q_n\circ \mu$ which are finite   
</ol> 
you see that, if your sentence $(\forall x)(\exists y)(w(x,y)=1)$ is true in all finite group, 
it must be true in all  $\Gamma_n$ and then in $\Gamma$, whence in all groups (same for the other sentence).