I'm looking for a theorem about quotient of quasi-isomorphic cdga's:

Let $A, B$ be two cdga's (commutative differential $\mathbb Z$-graded algebra) of nonpositive degrees, and $\mathfrak m \subset A, \mathfrak n \subset B$ be maximal ideals closed under differentials. Suppose $\phi: A \to B$ is a quasi-isomorphism of cdga's and $\phi(\mathfrak m) \subset \mathfrak n$. Is the induced map $\phi: A/ \mathfrak m^2 \to B/\mathfrak n^2$ still a quasi-isomorphism?

I guess more assumptions on $A,B, \mathfrak m, \mathfrak n$ might be needed. Any theorems of this type would be helpful.