Trying to understand an expression in Peano's Principia Arithmetices

[Nick O]

In the attached image, I am having trouble with point 66. I realize the notation is a bit archaic, so I'll explain as much as I can.

[(x,y)∈]a means "the set of solutions of the proposition a(x,y)." For example, the set of solutions to the proposition x < y.

-= means "is not equivalent to", and refers to two sets in each case here.

Λ means "the empty set".

The period is part of dot-parentheses notation. I'll tie everything together in parentheses in a little bit rather than explain it.

= means "if and only if" and links two propositions.

The next part is what bothers me.

I think [y∈]a is the set of solutions of a(x,y) in terms of x alone. But, what does this really mean? What would the resultant set be if a were the proposition x<y? What if it were x⇒y?

I suspect that this has some ties to lambda calculus, but that doesn't help me much since I have no real knowledge of it.

Now, all together, the tautology is:

([(x,y)∈]a -= Λ) = (([x∈]([y∈]a -= Λ)) -= Λ)

Can anyone help me understand the partial solution [y∈]a?

 

[Nick O]

Okay, I have been flipping through the pages of the discrete math book I'll be using next semester, and found a more intuitive (and probably more modern) way to understand this whole expression. It is simply this proposition:

∃x∃y(a(x,y)) ↔ ∃x(∃y(a(x,y)))