Theorem tsbi1

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

	Ref	Expression
Assertion	tsbi1	$$⊢\theta \to \left(\left(\neg \phi \vee \neg \psi \right)\vee \left(\phi \leftrightarrow \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		pm5.1	$$⊢\left(\phi \wedge \psi \right)\to \left(\phi \leftrightarrow \psi \right)$$
2	1	olcd	$$⊢\left(\phi \wedge \psi \right)\to \left(\left(\neg \phi \vee \neg \psi \right)\vee \left(\phi \leftrightarrow \psi \right)\right)$$
3		pm3.13	$$⊢\neg \left(\phi \wedge \psi \right)\to \left(\neg \phi \vee \neg \psi \right)$$
4	3	orcd	$$⊢\neg \left(\phi \wedge \psi \right)\to \left(\left(\neg \phi \vee \neg \psi \right)\vee \left(\phi \leftrightarrow \psi \right)\right)$$
5	2 4	pm2.61i	$$⊢\left(\left(\neg \phi \vee \neg \psi \right)\vee \left(\phi \leftrightarrow \psi \right)\right)$$
6	5	a1i	$$⊢\theta \to \left(\left(\neg \phi \vee \neg \psi \right)\vee \left(\phi \leftrightarrow \psi \right)\right)$$