The Wikipedia page on Stiefel-Whitney classes includes the following paragraph:
Over the Steenrod algebra, the Stiefel–Whitney classes of a smooth manifold (defined as the Stiefel–Whitney classes of its tangent bundle) are generated by those of the form $w_{2^i}$ . In particular, the Stiefel–Whitney classes satisfy the Wu formula, named for Wu Wenjun: $\text{Sq}^i(w_j)=\sum_{t=0}^i {j+t-i-1 \choose t} w_{i-t}w_{j+t}.$
This seems to suggest the first sentence follows from the second, but the reference provided (to May's A Concise Course in Algebraic Topology) doesn't mention this. Personally, I can't see the connection. Is there some well-known lemma I'm missing?