SQ-universal implies no nontrivial identity

Statement
Suppose $$G$$ is a SQ-universal group, i.e., a group such that every finite group is a subquotient of $$G$$. Then, $$G$$ is a group satisfying no nontrivial identity.

Applications

 * Finitary alternating group on infinite set implies no nontrivial identity

Proof
Given: A SQ-universal group $$G$$.

To prove: For every nontrivial word $$w$$, $$w$$ is not the identity for every choice of letters from $$G$$.

Proof: