Theorem wl-1xor

Description: In the recursive scheme

"(n+1)-xor" <-> if- ( ph , -. "n-xor" , "n-xor" )

we set n = 0 to formally arrive at an expression for "1-xor". The base case "0-xor" is replaced with F. , as a sequence of 0 inputs never has an odd number being part of it. (Contributed by Wolf Lammen, 11-May-2024)

	Ref	Expression
Assertion	wl-1xor	|- ( if- ( ps , -. F. , F. ) <-> ps )

Proof

Step	Hyp	Ref	Expression
1		tbtru	|- ( ps <-> ( ps <-> T. ) )
2	1	biimpi	|- ( ps -> ( ps <-> T. ) )
3		notfal	|- ( -. F. <-> T. )
4	2 3	bitr4di	|- ( ps -> ( ps <-> -. F. ) )
5		nbfal	|- ( -. ps <-> ( ps <-> F. ) )
6	5	biimpi	|- ( -. ps -> ( ps <-> F. ) )
7	4 6	casesifp	|- ( ps <-> if- ( ps , -. F. , F. ) )
8	7	bicomi	|- ( if- ( ps , -. F. , F. ) <-> ps )