Theorem tsbi2

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

	Ref	Expression
Assertion	tsbi2	$$⊢\theta \to \left(\left(\phi \vee \psi \right)\vee \left(\phi \leftrightarrow \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		pm5.21	$$⊢\left(\neg \phi \wedge \neg \psi \right)\to \left(\phi \leftrightarrow \psi \right)$$
2	1	olcd	$$⊢\left(\neg \phi \wedge \neg \psi \right)\to \left(\left(\phi \vee \psi \right)\vee \left(\phi \leftrightarrow \psi \right)\right)$$
3		pm4.57	$$⊢\neg \left(\neg \phi \wedge \neg \psi \right)\leftrightarrow \left(\phi \vee \psi \right)$$
4	3	biimpi	$$⊢\neg \left(\neg \phi \wedge \neg \psi \right)\to \left(\phi \vee \psi \right)$$
5	4	orcd	$$⊢\neg \left(\neg \phi \wedge \neg \psi \right)\to \left(\left(\phi \vee \psi \right)\vee \left(\phi \leftrightarrow \psi \right)\right)$$
6	2 5	pm2.61i	$$⊢\left(\left(\phi \vee \psi \right)\vee \left(\phi \leftrightarrow \psi \right)\right)$$
7	6	a1i	$$⊢\theta \to \left(\left(\phi \vee \psi \right)\vee \left(\phi \leftrightarrow \psi \right)\right)$$