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r - combine rows in data frame containing NA to make complete row

I know this is a duplicate Q but I can't seem to find the post again

Using the following data

df <- data.frame(A=c(1,1,2,2),B=c(NA,2,NA,4),C=c(3,NA,NA,5),D=c(NA,2,3,NA),E=c(5,NA,NA,4))

  A  B  C  D  E
  1 NA  3 NA  5
  1  2 NA  2 NA
  2 NA NA  3 NA
  2  4  5 NA  4

Grouping by A, I'd like the following output using a tidyverse solution

  A  B  C  D  E
  1  2  3  2  5
  2  4  5  3  4

I have many groups in A. I think I saw an answer using coalesce but am unsure how to get it work. I'd like a solution that works with characters as well. Thanks!

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I haven't figured out how to put the coalesce_by_column function inside the dplyr pipeline, but this works:

coalesce_by_column <- function(df) {
  return(coalesce(df[1], df[2]))
}

df %>%
  group_by(A) %>%
  summarise_all(coalesce_by_column)

##       A     B     C     D     E
##   <dbl> <dbl> <dbl> <dbl> <dbl>
## 1     1     2     3     2     5
## 2     2     4     5     3     4

Edit: include @Jon Harmon's solution for more than 2 members of a group

# Supply lists by splicing them into dots:
coalesce_by_column <- function(df) {
  return(dplyr::coalesce(!!! as.list(df)))
}

df %>%
  group_by(A) %>%
  summarise_all(coalesce_by_column)

#> # A tibble: 2 x 5
#>       A     B     C     D     E
#>   <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1     1     2     3     2     5
#> 2     2     4     5     3     4

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