Theorem wl-3xorfal

Description: If the first input is false, then triple xor is equivalent to the exclusive disjunction of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016) df-had redefined. (Revised by Wolf Lammen, 29-Apr-2024)

	Ref	Expression
Assertion	wl-3xorfal	⊢ ( ¬ 𝜑 → ( hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( 𝜓 ⊻ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		wl-df-3xor	⊢ ( hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ if- ( 𝜑 , ¬ ( 𝜓 ⊻ 𝜒 ) , ( 𝜓 ⊻ 𝜒 ) ) )
2		ifpfal	⊢ ( ¬ 𝜑 → ( if- ( 𝜑 , ¬ ( 𝜓 ⊻ 𝜒 ) , ( 𝜓 ⊻ 𝜒 ) ) ↔ ( 𝜓 ⊻ 𝜒 ) ) )
3	1 2	syl5bb	⊢ ( ¬ 𝜑 → ( hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( 𝜓 ⊻ 𝜒 ) ) )