Theorem xorbi12d

Description: Equality property for exclusive disjunction. (Contributed by Mario Carneiro, 4-Sep-2016)

	Ref	Expression
Hypotheses	xor12d.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
	xor12d.2	$$⊢\phi \to \left(\theta \leftrightarrow \tau \right)$$
Assertion	xorbi12d	$$⊢\phi \to \left(\left(\psi ⊻\theta \right)\leftrightarrow \left(\chi ⊻\tau \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		xor12d.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		xor12d.2	$$⊢\phi \to \left(\theta \leftrightarrow \tau \right)$$
3	1 2	bibi12d	$$⊢\phi \to \left(\left(\psi \leftrightarrow \theta \right)\leftrightarrow \left(\chi \leftrightarrow \tau \right)\right)$$
4	3	notbid	$$⊢\phi \to \left(\neg \left(\psi \leftrightarrow \theta \right)\leftrightarrow \neg \left(\chi \leftrightarrow \tau \right)\right)$$
5		df-xor	$$⊢\left(\psi ⊻\theta \right)\leftrightarrow \neg \left(\psi \leftrightarrow \theta \right)$$
6		df-xor	$$⊢\left(\chi ⊻\tau \right)\leftrightarrow \neg \left(\chi \leftrightarrow \tau \right)$$
7	4 5 6	3bitr4g	$$⊢\phi \to \left(\left(\psi ⊻\theta \right)\leftrightarrow \left(\chi ⊻\tau \right)\right)$$