I derived a relationship between sequences drawn with and without replacement for an application in genetics. The proof is easy enough, but I would rather find a source than provide a derivation of a well-known result. But can't seem to find it.

Simplified problem setup:
You have a stack of 52 card from which you draw uniformly at random *with* replacement until an arbitrary condition is met (say, you picked the ace of spades). This generates sequences with replacement.
The order in which cards are *first* picked defines a random sequence without replacement.

An alternative way of generating the same distribution of our original sequences *with* replacement is as follows:

First draw this "order of first sampling" by generating a random permutation of the cards (i.e., shuffle the deck).

Then you pick cards from the top of the deck and keep a stack of previously drawn cards. suppose that by the time you completed the n th draw, the "previously picked cards" deck has c(n) cards. We then pick randomly from the "previously picked" pile with probability c(n)/52, and from the top of the original deck with probability 1-c(n)/52

Again, I don't want a proof that the alternative way is equivalent to the original, I am just wondering whether people know a name or reference for this.