Theorem 3bior2fd

Description: A wff is equivalent to its threefold disjunction with double falsehood, analogous to biorf . (Contributed by Alexander van der Vekens, 8-Sep-2017)

	Ref	Expression
Hypotheses	3biorfd.1	$$⊢\phi \to \neg \theta$$
	3biorfd.2	$$⊢\phi \to \neg \chi$$
Assertion	3bior2fd	$$⊢\phi \to \left(\psi \leftrightarrow \left(\theta \vee \chi \vee \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		3biorfd.1	$$⊢\phi \to \neg \theta$$
2		3biorfd.2	$$⊢\phi \to \neg \chi$$
3		biorf	$$⊢\neg \chi \to \left(\psi \leftrightarrow \left(\chi \vee \psi \right)\right)$$
4	2 3	syl	$$⊢\phi \to \left(\psi \leftrightarrow \left(\chi \vee \psi \right)\right)$$
5	1	3bior1fd	$$⊢\phi \to \left(\left(\chi \vee \psi \right)\leftrightarrow \left(\theta \vee \chi \vee \psi \right)\right)$$
6	4 5	bitrd	$$⊢\phi \to \left(\psi \leftrightarrow \left(\theta \vee \chi \vee \psi \right)\right)$$