Theorem 3bior2fd 1476

Description: A wff is equivalent to its threefold disjunction with double falsehood, analogous to biorf 934. (Contributed by Alexander van der Vekens, 8-Sep-2017.)

Hypotheses

Ref	Expression
3biorfd.1	⊢ (𝜑 → ¬ 𝜃)
3biorfd.2	⊢ (𝜑 → ¬ 𝜒)

Assertion

Ref	Expression
3bior2fd	⊢ (𝜑 → (𝜓 ↔ (𝜃 ∨ 𝜒 ∨ 𝜓)))

Proof of Theorem 3bior2fd

Step	Hyp	Ref	Expression
1		3biorfd.2	. . 3 ⊢ (𝜑 → ¬ 𝜒)
2		biorf 934	. . 3 ⊢ (¬ 𝜒 → (𝜓 ↔ (𝜒 ∨ 𝜓)))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∨ 𝜓)))
4		3biorfd.1	. . 3 ⊢ (𝜑 → ¬ 𝜃)
5	4	3bior1fd 1474	. 2 ⊢ (𝜑 → ((𝜒 ∨ 𝜓) ↔ (𝜃 ∨ 𝜒 ∨ 𝜓)))
6	3, 5	bitrd 278	1 ⊢ (𝜑 → (𝜓 ↔ (𝜃 ∨ 𝜒 ∨ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∨ wo 844   ∨ w3o 1085

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 845  df-3or 1087

This theorem is referenced by: (None)