Subgroup generated by commutator of generators of free group on two generators is automorph-conjugate

Statement
Let $$F$$ be a free group on two generators, with $$x,y$$ being the generators. Let $$H$$ be the subgroup of $$F$$ generated by the commutator $$[x,y] = xyx^{-1}y^{-1}$$:

$$H = \langle [x,y] \rangle$$.

Then, $$H$$ is an automorph-conjugate subgroup of $$F$$.

Facts used

 * 1) uses::Automorph-conjugate iff conjugate to image under a generating set of automorphism group
 * 2) uses::Elementary Nielsen automorphisms generate the automorphism group of a finitely generated free group

Proof
Given: $$F$$ is a free group with freely generating set $$\{x,y\}$$. $$H = \langle [x,y] \rangle$$.

To prove: $$H$$ is automorph-conjugate in $$F$$.

Proof: By fact (2), the elementary Nielsen automorphisms of $$F$$ generate $$\operatorname{Aut}(F)$$. We use a modified version of this generating set to show that $$H$$ is automorph-conjugate in $$F$$ via fact (1):


 * Replacing $$x$$ by its inverse: $$\tau_x([x,y]) = [x^{-1},y] = x^{-1}yxy^{-1} = x^{-1}yxy^{-1} \cdot x^{-1}x = x^{-1}[y,x]x = x^{-1}[x,y]^{-1}x \in x^{-1}Hx$$.
 * Replacing $$y$$ by its inverse: $$\tau_y([x,y]) = [x,y^{-1}] = xy^{-1}x^{-1}y = y^{-1}[x,y]^{-1}y in y^{-1}Hy$$.
 * Swapping $$x$$ and $$y$$: $$\sigma([x,y]) = [y,x] = [x,y]^{-1}\in H$$.
 * Replacing $$x$$ by $$xy$$: $$\eta([x,y]) = xyyy^{-1}x^{-1}y^{-1} = xyx^{-1}y^{-1} = [x,y] \in H$$.