If I estimate the following regression on a large data set: $y = b_0x_0 + b_1x_1 + b_2x_2 + b_3x_3 + b_4x_4$ where $x_0, x_1,...x_4$ are all dummy variables indicating group membership, and I create two new variables, $x_5$ and $x_6$, that indicate membership such that $x_5$ indicates $x_0, x_1$, or $x_2$ being switched on, and $x_6$ indicates $x_3$ or $x_4$ being switched on, and then I estimate $y = b_5x_5 + b_6x_6$.
What would be the intuitive difference between testing the significance of $b_5$ and $b_6$ using t-stats vs. doing two separate F-tests, one on $b_0, b_1$, and $b_2$, and the other on $b_3$ and $b_4$? Is there any difference? Would the F-tests just be the t-stats squared in this case? Any help is appreciated.