Theorem 3bior1fd 1363

Description: A disjunction is equivalent to a threefold disjunction with single falsehood, analogous to biorf 745. (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 745	. . 3 ⊢ (¬ 𝜃 → ((𝜒 ∨ 𝜓) ↔ (𝜃 ∨ (𝜒 ∨ 𝜓))))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → ((𝜒 ∨ 𝜓) ↔ (𝜃 ∨ (𝜒 ∨ 𝜓))))
4		3orass 983	. 2 ⊢ ((𝜃 ∨ 𝜒 ∨ 𝜓) ↔ (𝜃 ∨ (𝜒 ∨ 𝜓)))
5	3, 4	bitr4di 198	1 ⊢ (𝜑 → ((𝜒 ∨ 𝜓) ↔ (𝜃 ∨ 𝜒 ∨ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 709   ∨ w3o 979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-3or 981

This theorem is referenced by:  3bior1fand  1364  3bior2fd  1365