As should be clear, I would like to know if it is true that a given commmutative square of spaces (i.e. simplicial sets) is a homotopy pullback iff the induced map on each homotopy fiber is a weak equivalence. More precisely, consider the following diagram: $$\begin{array}{c}&&&&& A& \longrightarrow & B\\ &&&&&\downarrow && \downarrow \\ &&&&& C &\longrightarrow & D \\ &&&&\nearrow & &\nearrow\\ &&&1 & \longrightarrow & 1 \end{array}$$ Suppose that, after having taken homotopy pullbacks on both sides, the resulting front face is a homotopy pullback (i.e. the top horizontal arrow [between homotopy fibers] is a weak equivalence), and this happens for any vertex of $C$. Is it true that the square involving $A,B,C,D$ is a homotopy pullback?

This seems to appear quite often but I couldn't find a precise reference with a proof.

Thanks in advance for your help.

As should be clear, I would like to know if it is true that a given commmutative square of spaces (i.e. simplicial sets) is a homotopy pullback iff the induced map on each homotopy fiber is a weak equivalence. More precisely, consider the following diagram: $$\begin{array}{c}&&&&& A& \longrightarrow & B\\ &&&&&\downarrow && \downarrow \\ &&&&& C &\longrightarrow & D \\ &&&&\nearrow & &\nearrow\\ &&&1 & \longrightarrow & 1 \end{array}$$ Suppose that, after having taken homotopy pullbacks on both sides, the resulting front face is a homotopy pullback (i.e. the top horizontal arrow [between homotopy fibers] is a weak equivalence), and that this happens for any vertex of $C$. Is it true that the square involving $A,B,C,D$ is a homotopy pullback?

This seems to appear quite often but I couldn't find a precise reference with a proof.

Thanks in advance for your help.