Theorem 3bior1fd

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

	Ref	Expression
Hypothesis	3biorfd.1	⊢ ( 𝜑 → ¬ 𝜃 )
Assertion	3bior1fd	⊢ ( 𝜑 → ( ( 𝜒 ∨ 𝜓 ) ↔ ( 𝜃 ∨ 𝜒 ∨ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		3biorfd.1	⊢ ( 𝜑 → ¬ 𝜃 )
2		biorf	⊢ ( ¬ 𝜃 → ( ( 𝜒 ∨ 𝜓 ) ↔ ( 𝜃 ∨ ( 𝜒 ∨ 𝜓 ) ) ) )
3	1 2	syl	⊢ ( 𝜑 → ( ( 𝜒 ∨ 𝜓 ) ↔ ( 𝜃 ∨ ( 𝜒 ∨ 𝜓 ) ) ) )
4		3orass	⊢ ( ( 𝜃 ∨ 𝜒 ∨ 𝜓 ) ↔ ( 𝜃 ∨ ( 𝜒 ∨ 𝜓 ) ) )
5	3 4	bitr4di	⊢ ( 𝜑 → ( ( 𝜒 ∨ 𝜓 ) ↔ ( 𝜃 ∨ 𝜒 ∨ 𝜓 ) ) )