"Back your life it is". What does it mean?

The following excerpt is from the story 'Adolf' by DH Lawrence, in which there is the expression 'Back your life it is'. What does it mean? I googled to find a definition and looked up in many dictionaries, but all in vain. Is it an idiomatic expression or a fixed phrase—slang or dialect?

When morning came, and it was light, I went downstairs. Opening the scullery door I heard a slight scuffle. Then I saw dabbles of milk all over the floor and tiny rabbit-droppings in the saucers. And there the miscreant, the tips of his ears showing behind a pair of boots. I peeped at him. He sat bright-eyed and askance, twitching his nose and looking at me while not looking at me.

He was alive — very much alive. But still we were afraid to trespass much on his confidence.

"Father!" My father was arrested at the door. "Father, the rabbit's alive."

"Back your life it is," said my father.

"Mind how you go in."

Solution 1:

I think there is a simple answer to this. There is an idiomatic exclamation in English:

When I want to re-enforce a factual claim, I might say this:

"That rabbit will still be in the scullery tomorrow, you can bet on it." Or I could say, "... you can bet your life on it" and gain a greater impression of certainty.

Another word for 'bet' is 'back'. You can bet on a horse or you can back a horse.

So I think it likely is that the father is saying that he is strongly shares the certainty. If I am right, the verb "back" is imperative. To the parent is telling the child he can be quite sure the rabbit is still alive: so sure that he can back (i.e bet) his life it is.