Theorem 3bior1fand

Description: A disjunction is equivalent to a threefold disjunction with single falsehood of a conjunction. (Contributed by Alexander van der Vekens, 8-Sep-2017)

	Ref	Expression
Hypothesis	3biorfd.1	$$⊢\phi \to \neg \theta$$
Assertion	3bior1fand	$$⊢\phi \to \left(\left(\chi \vee \psi \right)\leftrightarrow \left(\left(\theta \wedge \tau \right)\vee \chi \vee \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		3biorfd.1	$$⊢\phi \to \neg \theta$$
2	1	intnanrd	$$⊢\phi \to \neg \left(\theta \wedge \tau \right)$$
3	2	3bior1fd	$$⊢\phi \to \left(\left(\chi \vee \psi \right)\leftrightarrow \left(\left(\theta \wedge \tau \right)\vee \chi \vee \psi \right)\right)$$