I have started to learn about the mathematics behind ebits and I have a question. Assume $\color{red}{\text{Alice}}$ and $\color{blue}{\text{Bob}}$ share the following ebit: $\begin{align}\vert\Phi^+ \rangle= \dfrac{\vert\color{red}{0}\color{blue}{0}\rangle^{\color{red}{A}\color{blue}{B}} + \vert\color{red}{1}\color{blue}{1}\rangle^{\color{red}{A}\color{blue}{B}}}{\sqrt{2}} .\end{align}$
What would happen if $\color{red}{\text{Alice}}$ performs Pauli-$X$ gate on their portion of the ebit?
Here is my answer but I am not sure if it is correct:
The $\color{red}{0}$ and $\color{red}{1}$ will swap and end up with: $\color{red}{\text{Alice}}$ and $\color{blue}{\text{Bob}}$ share the following ebit: $\begin{align}\vert\Phi^+ \rangle= \dfrac{\vert\color{red}{1}\color{blue}{0}\rangle^{\color{red}{A}\color{blue}{B}} + \vert\color{red}{0}\color{blue}{1}\rangle^{\color{red}{A}\color{blue}{B}}}{\sqrt{2}} .\end{align}$
Any help is appreciated!