Theorem anddi 1010

Description: Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)

Assertion

Ref	Expression
anddi	⊢ (((𝜑 ∨ 𝜓) ∧ (𝜒 ∨ 𝜃)) ↔ (((𝜑 ∧ 𝜒) ∨ (𝜑 ∧ 𝜃)) ∨ ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))))

Proof of Theorem anddi

Step	Hyp	Ref	Expression
1		andir 1008	. 2 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜒 ∨ 𝜃)) ↔ ((𝜑 ∧ (𝜒 ∨ 𝜃)) ∨ (𝜓 ∧ (𝜒 ∨ 𝜃))))
2		andi 1007	. . 3 ⊢ ((𝜑 ∧ (𝜒 ∨ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∨ (𝜑 ∧ 𝜃)))
3		andi 1007	. . 3 ⊢ ((𝜓 ∧ (𝜒 ∨ 𝜃)) ↔ ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃)))
4	2, 3	orbi12i 914	. 2 ⊢ (((𝜑 ∧ (𝜒 ∨ 𝜃)) ∨ (𝜓 ∧ (𝜒 ∨ 𝜃))) ↔ (((𝜑 ∧ 𝜒) ∨ (𝜑 ∧ 𝜃)) ∨ ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))))
5	1, 4	bitri 278	1 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜒 ∨ 𝜃)) ↔ (((𝜑 ∧ 𝜒) ∨ (𝜑 ∧ 𝜃)) ∨ ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))))

Syntax hints:   ↔ wb 209   ∧ wa 399   ∨ wo 846

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 210  df-an 400  df-or 847

This theorem is referenced by:  prnebg  4738  funun  6379  disjxpin  30493  icoreclin  35140  undif3VD  42024