The second form of SCH implies the first

Assume the second form of the singular cardinals hypothesis: for every singular cardinal $\lambda$, if $2^{\operatorname{cf}\lambda}<\lambda$ then $\lambda^{\operatorname{cf}\lambda}=\lambda^+$. Then the first form holds: for every singular strong limit cardinal $\kappa$,

The source states that the second form is strictly stronger; this theorem is the implication half of that claim (strictness is a consistency statement and is not formalized).