Change the level of a factor based on the level of another factor

I have a data set with many variables, two of which called "animal" and "plant". Both variable are factors, and both are binary, i.e. they are either a text value, or NA.

For example:

animal <- c(NA, NA, "cat", "cat", NA)
plant  <- c("ivy", NA, "ivy", NA, NA)
value  <- c(1:5)
df     <- data.frame(animal, plant, value)

> df
  animal plant value
1   <NA>   ivy     1
2   <NA>  <NA>     2
3    cat   ivy     3
4    cat  <NA>     4
5   <NA>  <NA>     5

When the value of plant is "ivy" and the value of animal is "cat", I want to change the value of plant to NA (i,e, the two things can not be true and the animal value takes priority. I don't any changes in my other variables

I've tried the following but get an error message:

df <- df %>% if (isTRUE(animal == "cat")) {plant==NA}

Error in if (.) isTRUE(animal == "cat") else { : 
  argument is not interpretable as logical
In addition: Warning message:
In if (.) isTRUE(animal == "cat") else { :
  the condition has length > 1 and only the first element will be used

My goal output is:

> df
  animal plant value
1   <NA>   ivy     1
2   <NA>  <NA>     2
3    cat  <NA>     3
4    cat  <NA>     4
5   <NA>  <NA>     5

I would really appreciate any help. I'm sure there is a really simple way of doing this, maybe I can't see the wood for the trees.