Consider a random walk on the integer lattice of the positive quadrant in two dimensions. If at any step the process is at $(m,n)$, it moves at the next step to $(m+1,n)$ or $(m,n+1)$ with probability $\frac{1}{2}$ each. Let the process start at $(0,0)$. Let $\Gamma$ be any curve connecting neighboring lattice points extending from the Y-axis to the X-axis in the first quadrant. Show that $E[Y_1]=E[Y_2]$ where $Y_1$ and $Y_2$ denote the number of steps to the right and up, respectively, before hitting the boundary $\Gamma$.
Intuitively, I know that $E[Y_1]=E[Y_2]= \frac{1}{2}E[T]$ where $T$ is the number of steps it takes to reach the boundary. On the random walk, since the probabilities of going right and up are $\frac{1}{2}$, you can expect it to follow along the line $y=x$ before it hits the boundary. This means that the distance up is equal to the distance to the right. We either have that the number of steps up is equal to the number of steps to the right, or the number of steps for either is one step more than the other. There is an equal probability of this happening, so the expected number of steps up is equal to the expected number of steps to the right.
I know this is in no way an acceptable proof because it is not rigorous at all. Could I get some help creating a formal proof for this? Thank you.