Theorem wl-df3xor2

Description: Alternative definition of wl-df-3xor , using triple exclusive disjunction, or XOR3. You can add more input by appending each one with a \/_ . Copy of hadass . (Contributed by Mario Carneiro, 4-Sep-2016) df-had redefined. (Revised by Wolf Lammen, 1-May-2024)

	Ref	Expression
Assertion	wl-df3xor2	|- ( hadd ( ph , ps , ch ) <-> ( ph \/_ ( ps \/_ ch ) ) )

Proof

Step	Hyp	Ref	Expression
1		ifpn	|- ( if- ( ph , -. ( ps \/_ ch ) , ( ps \/_ ch ) ) <-> if- ( -. ph , ( ps \/_ ch ) , -. ( ps \/_ ch ) ) )
2		wl-df-3xor	|- ( hadd ( ph , ps , ch ) <-> if- ( ph , -. ( ps \/_ ch ) , ( ps \/_ ch ) ) )
3		df-xor	|- ( ( ph \/_ ( ps \/_ ch ) ) <-> -. ( ph <-> ( ps \/_ ch ) ) )
4		nbbn	|- ( ( -. ph <-> ( ps \/_ ch ) ) <-> -. ( ph <-> ( ps \/_ ch ) ) )
5		ifpdfbi	|- ( ( -. ph <-> ( ps \/_ ch ) ) <-> if- ( -. ph , ( ps \/_ ch ) , -. ( ps \/_ ch ) ) )
6	3 4 5	3bitr2i	|- ( ( ph \/_ ( ps \/_ ch ) ) <-> if- ( -. ph , ( ps \/_ ch ) , -. ( ps \/_ ch ) ) )
7	1 2 6	3bitr4i	|- ( hadd ( ph , ps , ch ) <-> ( ph \/_ ( ps \/_ ch ) ) )