Theorem orordir 769

Description: Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)

Assertion

Ref	Expression
orordir	⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ ((𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜒)))

Proof of Theorem orordir

Step	Hyp	Ref	Expression
1		oridm 752	. . 3 ⊢ ((𝜒 ∨ 𝜒) ↔ 𝜒)
2	1	orbi2i 757	. 2 ⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒))
3		or4 766	. 2 ⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜒)) ↔ ((𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜒)))
4	2, 3	bitr3i 185	1 ⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ ((𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜒)))

Syntax hints:   ↔ wb 104   ∨ wo 703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  elznn0  9227