Can "$x(\lambda y.P\;)z$" be $\beta$-reduced?
The expression $x(\lambda y.P)z$ is not $\beta$-reducible (unless there is a redex inside $P$, that is, but no such reduction sequence can escape outside the $\lambda$). The reduction you mention is invalid even in $\lambda\eta$-calculus. The only explanation I can imagine is that the expression in your derivation was misbracketed, and should actually read $x((\lambda y.P)z)$.