Theorem tsxo1

Description: A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018)

	Ref	Expression
Assertion	tsxo1	$$⊢\theta \to \left(\left(\neg \phi \vee \neg \psi \right)\vee \neg \left(\phi ⊻\psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		tsbi1	$$⊢\theta \to \left(\left(\neg \phi \vee \neg \psi \right)\vee \left(\phi \leftrightarrow \psi \right)\right)$$
2		xnor	$$⊢\left(\phi \leftrightarrow \psi \right)\leftrightarrow \neg \left(\phi ⊻\psi \right)$$
3	2	orbi2i	$$⊢\left(\left(\neg \phi \vee \neg \psi \right)\vee \left(\phi \leftrightarrow \psi \right)\right)\leftrightarrow \left(\left(\neg \phi \vee \neg \psi \right)\vee \neg \left(\phi ⊻\psi \right)\right)$$
4	1 3	sylib	$$⊢\theta \to \left(\left(\neg \phi \vee \neg \psi \right)\vee \neg \left(\phi ⊻\psi \right)\right)$$