Theorem ts3an1

Description: A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018)

	Ref	Expression
Assertion	ts3an1	$$⊢\theta \to \left(\left(\neg \left(\phi \wedge \psi \right)\vee \neg \chi \right)\vee \left(\phi \wedge \psi \wedge \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		tsan1	$$⊢\theta \to \left(\left(\neg \left(\phi \wedge \psi \right)\vee \neg \chi \right)\vee \left(\left(\phi \wedge \psi \right)\wedge \chi \right)\right)$$
2		df-3an	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\leftrightarrow \left(\left(\phi \wedge \psi \right)\wedge \chi \right)$$
3	2	orbi2i	$$⊢\left(\left(\neg \left(\phi \wedge \psi \right)\vee \neg \chi \right)\vee \left(\phi \wedge \psi \wedge \chi \right)\right)\leftrightarrow \left(\left(\neg \left(\phi \wedge \psi \right)\vee \neg \chi \right)\vee \left(\left(\phi \wedge \psi \right)\wedge \chi \right)\right)$$
4	1 3	sylibr	$$⊢\theta \to \left(\left(\neg \left(\phi \wedge \psi \right)\vee \neg \chi \right)\vee \left(\phi \wedge \psi \wedge \chi \right)\right)$$