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 ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( 𝜑 ⊻ ( 𝜓 ⊻ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ifpn	⊢ ( if- ( 𝜑 , ¬ ( 𝜓 ⊻ 𝜒 ) , ( 𝜓 ⊻ 𝜒 ) ) ↔ if- ( ¬ 𝜑 , ( 𝜓 ⊻ 𝜒 ) , ¬ ( 𝜓 ⊻ 𝜒 ) ) )
2		wl-df-3xor	⊢ ( hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ if- ( 𝜑 , ¬ ( 𝜓 ⊻ 𝜒 ) , ( 𝜓 ⊻ 𝜒 ) ) )
3		df-xor	⊢ ( ( 𝜑 ⊻ ( 𝜓 ⊻ 𝜒 ) ) ↔ ¬ ( 𝜑 ↔ ( 𝜓 ⊻ 𝜒 ) ) )
4		nbbn	⊢ ( ( ¬ 𝜑 ↔ ( 𝜓 ⊻ 𝜒 ) ) ↔ ¬ ( 𝜑 ↔ ( 𝜓 ⊻ 𝜒 ) ) )
5		ifpdfbi	⊢ ( ( ¬ 𝜑 ↔ ( 𝜓 ⊻ 𝜒 ) ) ↔ if- ( ¬ 𝜑 , ( 𝜓 ⊻ 𝜒 ) , ¬ ( 𝜓 ⊻ 𝜒 ) ) )
6	3 4 5	3bitr2i	⊢ ( ( 𝜑 ⊻ ( 𝜓 ⊻ 𝜒 ) ) ↔ if- ( ¬ 𝜑 , ( 𝜓 ⊻ 𝜒 ) , ¬ ( 𝜓 ⊻ 𝜒 ) ) )
7	1 2 6	3bitr4i	⊢ ( hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( 𝜑 ⊻ ( 𝜓 ⊻ 𝜒 ) ) )