Theorem tsxo4

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

	Ref	Expression
Assertion	tsxo4	$$⊢\theta \to \left(\left(\neg \phi \vee \psi \right)\vee \left(\phi ⊻\psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		tsbi4	$$⊢\theta \to \left(\left(\neg \phi \vee \psi \right)\vee \neg \left(\phi \leftrightarrow \psi \right)\right)$$
2		df-xor	$$⊢\left(\phi ⊻\psi \right)\leftrightarrow \neg \left(\phi \leftrightarrow \psi \right)$$
3	2	bicomi	$$⊢\neg \left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\phi ⊻\psi \right)$$
4	3	orbi2i	$$⊢\left(\left(\neg \phi \vee \psi \right)\vee \neg \left(\phi \leftrightarrow \psi \right)\right)\leftrightarrow \left(\left(\neg \phi \vee \psi \right)\vee \left(\phi ⊻\psi \right)\right)$$
5	1 4	sylib	$$⊢\theta \to \left(\left(\neg \phi \vee \psi \right)\vee \left(\phi ⊻\psi \right)\right)$$