R - handling NA in ifelse statements

This question is slightly similar to this question with a more theoretical component.

Given df below:

varA <- c(1,0,0,NA,NA)
varB <- c(NA,NA,NA,1,0)
df <- data.frame(varA, varB)

varA varB
1    NA
0    NA
0    NA
NA    1
NA    0

What's the most elegant method to generate var (with consideration given to NA) which combines the information from varA and varB?

varA varB var
1    NA   1
0    NA   0
0    NA   0
NA    1   1
NA    0   0

My approach, right now, is as follows:

df$var[df$varA == 1 | df$varB == 1] <- 1
df$var[df$varA == 0 | df$varB == 0] <- 0

As a side question, how does R handle NA in ifelse statements? For example, if I write the following code, it does not produce the output I intended.

df$var <- ifelse(df$varA == 1 | df$varB == 1, 1, 
            ifelse(df$varA == 0 | df$varB == 0, 0, NA)