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	⊢ ( 𝜑 → ¬ 𝜃 )
	3biorfd.2	⊢ ( 𝜑 → ¬ 𝜒 )
Assertion	3bior2fd	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜃 ∨ 𝜒 ∨ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		3biorfd.1	⊢ ( 𝜑 → ¬ 𝜃 )
2		3biorfd.2	⊢ ( 𝜑 → ¬ 𝜒 )
3		biorf	⊢ ( ¬ 𝜒 → ( 𝜓 ↔ ( 𝜒 ∨ 𝜓 ) ) )
4	2 3	syl	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜒 ∨ 𝜓 ) ) )
5	1	3bior1fd	⊢ ( 𝜑 → ( ( 𝜒 ∨ 𝜓 ) ↔ ( 𝜃 ∨ 𝜒 ∨ 𝜓 ) ) )
6	4 5	bitrd	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜃 ∨ 𝜒 ∨ 𝜓 ) ) )