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	$$⊢\neg \phi \to \left(hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\psi ⊻\chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		wl-df-3xor	$$⊢hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow if-\left(\phi ,\neg \left(\psi ⊻\chi \right),\left(\psi ⊻\chi \right)\right)$$
2		ifpfal	$$⊢\neg \phi \to \left(if-\left(\phi ,\neg \left(\psi ⊻\chi \right),\left(\psi ⊻\chi \right)\right)\leftrightarrow \left(\psi ⊻\chi \right)\right)$$
3	1 2	syl5bb	$$⊢\neg \phi \to \left(hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\psi ⊻\chi \right)\right)$$