Make a new dataframe of duplicated rows by column names from boolean columns?

I have a boolean table of four columns in a dataframe with some other fields e.g.

  A B C D X
  ---------
1 T T F T 0.1
2 F T F F 0.2
3 F F T T 0.3

There are more than one field X but let's just use X for this example.

How can I make a new table of duplicated rows by column names?

  Type  X
  -------
1 A   0.1
1 B   0.1
1 D   0.1
2 B   0.2
3 C   0.3
3 D   0.3

library(tidyr)
library(dplyr)

pivot_longer(df, cols=A:D, names_to="Type") |>
   filter(value) |>
   select(-value)

# A tibble: 6 × 3
     id     X Type 
  <dbl> <dbl> <chr>
1     1   0.1 A    
2     1   0.1 B    
3     1   0.1 D    
4     2   0.2 B    
5     3   0.3 C    
6     3   0.3 D

Data:

df <- structure(list(id = c(1, 2, 3), A = c(TRUE, FALSE, FALSE), B = c(TRUE, 
TRUE, FALSE), C = c(FALSE, FALSE, TRUE), D = c(TRUE, FALSE, TRUE
), X = c(0.1, 0.2, 0.3)), class = c("tbl_df", "tbl", "data.frame"
), row.names = c(NA, -3L))