Theorem tsna3

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

	Ref	Expression
Assertion	tsna3	$$⊢\theta \to \left(\psi \vee \left(\phi ⊼\psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		tsan3	$$⊢\theta \to \left(\psi \vee \neg \left(\phi \wedge \psi \right)\right)$$
2		df-nan	$$⊢\left(\phi ⊼\psi \right)\leftrightarrow \neg \left(\phi \wedge \psi \right)$$
3	2	orbi2i	$$⊢\left(\psi \vee \left(\phi ⊼\psi \right)\right)\leftrightarrow \left(\psi \vee \neg \left(\phi \wedge \psi \right)\right)$$
4	1 3	sylibr	$$⊢\theta \to \left(\psi \vee \left(\phi ⊼\psi \right)\right)$$