Theorem tsbi4

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

	Ref	Expression
Assertion	tsbi4	$$⊢\theta \to \left(\left(\neg \phi \vee \psi \right)\vee \neg \left(\phi \leftrightarrow \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		tsbi3	$$⊢\theta \to \left(\left(\psi \vee \neg \phi \right)\vee \neg \left(\psi \leftrightarrow \phi \right)\right)$$
2		orcom	$$⊢\left(\psi \vee \neg \phi \right)\leftrightarrow \left(\neg \phi \vee \psi \right)$$
3		bicom	$$⊢\left(\psi \leftrightarrow \phi \right)\leftrightarrow \left(\phi \leftrightarrow \psi \right)$$
4	3	notbii	$$⊢\neg \left(\psi \leftrightarrow \phi \right)\leftrightarrow \neg \left(\phi \leftrightarrow \psi \right)$$
5	2 4	orbi12i	$$⊢\left(\left(\psi \vee \neg \phi \right)\vee \neg \left(\psi \leftrightarrow \phi \right)\right)\leftrightarrow \left(\left(\neg \phi \vee \psi \right)\vee \neg \left(\phi \leftrightarrow \psi \right)\right)$$
6	1 5	sylib	$$⊢\theta \to \left(\left(\neg \phi \vee \psi \right)\vee \neg \left(\phi \leftrightarrow \psi \right)\right)$$