Theorem xorbi12i

Description: Equality property for exclusive disjunction. (Contributed by Mario Carneiro, 4-Sep-2016) (Proof shortened by Wolf Lammen, 21-Apr-2024)

	Ref	Expression
Hypotheses	xorbi12.1	$$⊢\phi \leftrightarrow \psi$$
	xorbi12.2	$$⊢\chi \leftrightarrow \theta$$
Assertion	xorbi12i	$$⊢\left(\phi ⊻\chi \right)\leftrightarrow \left(\psi ⊻\theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		xorbi12.1	$$⊢\phi \leftrightarrow \psi$$
2		xorbi12.2	$$⊢\chi \leftrightarrow \theta$$
3		df-xor	$$⊢\left(\phi ⊻\chi \right)\leftrightarrow \neg \left(\phi \leftrightarrow \chi \right)$$
4	1 2	bibi12i	$$⊢\left(\phi \leftrightarrow \chi \right)\leftrightarrow \left(\psi \leftrightarrow \theta \right)$$
5	3 4	xchbinx	$$⊢\left(\phi ⊻\chi \right)\leftrightarrow \neg \left(\psi \leftrightarrow \theta \right)$$
6		df-xor	$$⊢\left(\psi ⊻\theta \right)\leftrightarrow \neg \left(\psi \leftrightarrow \theta \right)$$
7	5 6	bitr4i	$$⊢\left(\phi ⊻\chi \right)\leftrightarrow \left(\psi ⊻\theta \right)$$