I'm reading the book of Andrei Moroianu, "Lectures on kahler geometry" and at the page 67 is this lemma: 
And when it comes time for the proof he sais that $(2)$ and $(3)$ are "tautological". I'm kinda new to the subject and my question is why $(2)$ and $(3)$ are equivalent?
For $\phi_t$ the flow of $X$, we have that $$ \mathcal{L}_XJ(Y) = \mathcal{L}_X(JY) -J\mathcal{L}_X(Y) = \lim\limits_{t\to 0}\frac{(\phi_{-t})_\ast JY - JY}{t} - J\lim\limits_{t\to 0}\frac{(\phi_{-t})_\ast Y - Y}{t} = {\lim\limits_{t\to 0}\frac{(\phi_{-t})_\ast JY - JY}{t} - \lim\limits_{t\to 0}\frac{J(\phi_{-t})_\ast Y - JY}{t} = \lim\limits_{t\to 0}\frac{((\phi_{-t})_\ast J -J(\phi_{-t})_\ast) Y }{t} = \lim\limits_{t\to 0}\frac{[(\phi_{-t})_\ast, J]Y }{t} } $$ If $\phi_t$ is holomorphic for each $t$ then $[(\phi_{-t})_\ast, J]=0$ (by definition) which implies $\mathcal{L}_XJ=0$.
Now suppose that $\mathcal{L}_XJ=0$. Then $[(\phi_{t})_\ast, J]$ is constant for the variabble $t$ and we have $$ [(\phi_{t})_\ast, J] = [(\phi_{0})_\ast, J] = [Id,J]=0 $$ which implies that $\phi_{t}$ is holomorphic.