Theorem 3bior1fand 1477

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

Hypothesis

Ref	Expression
3biorfd.1	⊢ (𝜑 → ¬ 𝜃)

Assertion

Ref	Expression
3bior1fand	⊢ (𝜑 → ((𝜒 ∨ 𝜓) ↔ ((𝜃 ∧ 𝜏) ∨ 𝜒 ∨ 𝜓)))

Proof of Theorem 3bior1fand

Step	Hyp	Ref	Expression
1		3biorfd.1	. . 3 ⊢ (𝜑 → ¬ 𝜃)
2	1	intnanrd 489	. 2 ⊢ (𝜑 → ¬ (𝜃 ∧ 𝜏))
3	2	3bior1fd 1476	1 ⊢ (𝜑 → ((𝜒 ∨ 𝜓) ↔ ((𝜃 ∧ 𝜏) ∨ 𝜒 ∨ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∨ 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 207  df-an 396  df-or 848  df-3or 1087

This theorem is referenced by: (None)