Theorem anddi 767

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 765	. 2 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜒 ∨ 𝜃)) ↔ ((𝜑 ∧ (𝜒 ∨ 𝜃)) ∨ (𝜓 ∧ (𝜒 ∨ 𝜃))))
2		andi 764	. . 3 ⊢ ((𝜑 ∧ (𝜒 ∨ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∨ (𝜑 ∧ 𝜃)))
3		andi 764	. . 3 ⊢ ((𝜓 ∧ (𝜒 ∨ 𝜃)) ↔ ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃)))
4	2, 3	orbi12i 713	. 2 ⊢ (((𝜑 ∧ (𝜒 ∨ 𝜃)) ∨ (𝜓 ∧ (𝜒 ∨ 𝜃))) ↔ (((𝜑 ∧ 𝜒) ∨ (𝜑 ∧ 𝜃)) ∨ ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))))
5	1, 4	bitri 182	1 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜒 ∨ 𝜃)) ↔ (((𝜑 ∧ 𝜒) ∨ (𝜑 ∧ 𝜃)) ∨ ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))))

Syntax hints:   ∧ wa 102   ↔ 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:  funun  4964  acexmidlemcase  5527  nnm00  6125