T-test on indicator for a group of betas vs. F-test on the same group

Question

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.

Answer 1

$x_5 = 1-(1-x_0) (1-x_1) (1-x_2)$; $x_6= 1- (1-x_3)(1-x_4)$

As such,

$$b_5 x_5 = b_5 [1-(1-x_0) (1-x_1) (1-x_2)] \\ = b_5 x0 +b_5 x_1 + b5 x_2-b_5x_0x_1 -b_5x_0x_2 -b_5x_1x_2 +b_5x_0x_1x_2$$

That is not the same as $b_0x_0+b_1x_1+b_2x_2$.

Similarly with $x_6$ in terms of $x_3$ and $x_4$.