Theorem 3bior1fd 1473

Description: A disjunction is equivalent to a threefold disjunction with single falsehood, analogous to biorf 933. (Contributed by Alexander van der Vekens, 8-Sep-2017.)

Hypothesis

Ref	Expression
3biorfd.1	⊢ (𝜑 → ¬ 𝜃)

Assertion

Ref	Expression
3bior1fd	⊢ (𝜑 → ((𝜒 ∨ 𝜓) ↔ (𝜃 ∨ 𝜒 ∨ 𝜓)))

Proof of Theorem 3bior1fd

Step	Hyp	Ref	Expression
1		3biorfd.1	. . 3 ⊢ (𝜑 → ¬ 𝜃)
2		biorf 933	. . 3 ⊢ (¬ 𝜃 → ((𝜒 ∨ 𝜓) ↔ (𝜃 ∨ (𝜒 ∨ 𝜓))))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → ((𝜒 ∨ 𝜓) ↔ (𝜃 ∨ (𝜒 ∨ 𝜓))))
4		3orass 1088	. 2 ⊢ ((𝜃 ∨ 𝜒 ∨ 𝜓) ↔ (𝜃 ∨ (𝜒 ∨ 𝜓)))
5	3, 4	bitr4di 288	1 ⊢ (𝜑 → ((𝜒 ∨ 𝜓) ↔ (𝜃 ∨ 𝜒 ∨ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∨ wo 843   ∨ w3o 1084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-or 844  df-3or 1086

This theorem is referenced by:  3bior1fand  1474  3bior2fd  1475  nb3grprlem2  27651