Theorem 3bior2fd 1508

Description: A wff is equivalent to its threefold disjunction with double falsehood, analogous to biorf 950. (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 950	. . 3 ⊢ (¬ 𝜒 → (𝜓 ↔ (𝜒 ∨ 𝜓)))
3	1, 2	syl 18	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∨ 𝜓)))
4		3biorfd.1	. . 3 ⊢ (𝜑 → ¬ 𝜃)
5	4	3bior1fd 1506	. 2 ⊢ (𝜑 → ((𝜒 ∨ 𝜓) ↔ (𝜃 ∨ 𝜒 ∨ 𝜓)))
6	3, 5	bitrd 282	1 ⊢ (𝜑 → (𝜓 ↔ (𝜃 ∨ 𝜒 ∨ 𝜓)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 861   ∨ w3o 1102

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-or 862  df-3or 1104

This theorem is used by:  usgrexmpl2nb0  49129  usgrexmpl2nb3  49132