Memoryless conditional expectation of shifted function exponential

Related to this, is the following valid:

\begin{align} E[f(X-t) \mid X>t] = \int f(y-t) f_{X|X>t}(y) dy = \int f(x) f_{X|X>t}(x+t)dx = \int f(x) f_X(x) dx = E[f(X)] \end{align} where I make the substitution $$x=y-t$$ and then use memorylessness?