Uniqueness of symmetric basis in quasi-Banach spaces

We show that if X is a nonlocally convex natural quasi-Banach space with symmetric basis whose Banach envelope is isomorphic to ℓ1, then all symmetric bases of X are equivalent. The scope of this result is quite ample since the Banach envelopes of natural quasi-Banach spaces with basis always exhibit an ℓ1-like behavior, in the sense that they contain copies of 's uniformly complemented.