Theorem wl-3xorbi

Description: Triple xor can be replaced with a triple biconditional. Unlike \/_ , you cannot add more inputs by simply stacking up more biconditionals, and still express an "odd number of inputs". (Contributed by Wolf Lammen, 24-Apr-2024)

	Ref	Expression
Assertion	wl-3xorbi	|- ( hadd ( ph , ps , ch ) <-> ( ph <-> ( ps <-> ch ) ) )

Proof

Step	Hyp	Ref	Expression
1		wl-df3xor2	|- ( hadd ( ph , ps , ch ) <-> ( ph \/_ ( ps \/_ ch ) ) )
2		df-xor	|- ( ( ph \/_ ( ps \/_ ch ) ) <-> -. ( ph <-> ( ps \/_ ch ) ) )
3		xor3	|- ( -. ( ph <-> ( ps \/_ ch ) ) <-> ( ph <-> -. ( ps \/_ ch ) ) )
4		xnor	|- ( ( ps <-> ch ) <-> -. ( ps \/_ ch ) )
5	4	bibi2i	|- ( ( ph <-> ( ps <-> ch ) ) <-> ( ph <-> -. ( ps \/_ ch ) ) )
6	3 5	bitr4i	|- ( -. ( ph <-> ( ps \/_ ch ) ) <-> ( ph <-> ( ps <-> ch ) ) )
7	1 2 6	3bitri	|- ( hadd ( ph , ps , ch ) <-> ( ph <-> ( ps <-> ch ) ) )