Theorem orordi 722

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

Assertion

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

Proof of Theorem orordi

Step	Hyp	Ref	Expression
1		oridm 706	. . 3 ⊢ ((𝜑 ∨ 𝜑) ↔ 𝜑)
2	1	orbi1i 712	. 2 ⊢ (((𝜑 ∨ 𝜑) ∨ (𝜓 ∨ 𝜒)) ↔ (𝜑 ∨ (𝜓 ∨ 𝜒)))
3		or4 720	. 2 ⊢ (((𝜑 ∨ 𝜑) ∨ (𝜓 ∨ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ∨ (𝜑 ∨ 𝜒)))
4	2, 3	bitr3i 184	1 ⊢ ((𝜑 ∨ (𝜓 ∨ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ∨ (𝜑 ∨ 𝜒)))

Syntax hints:   ↔ wb 103   ∨ wo 661

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662

This theorem depends on definitions:  df-bi 115

This theorem is referenced by: (None)