Theorem ordir 807

Description: Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)

Assertion

Ref	Expression
ordir	⊢ (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ ((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)))

Proof of Theorem ordir

Step	Hyp	Ref	Expression
1		ordi 806	. 2 ⊢ ((𝜒 ∨ (𝜑 ∧ 𝜓)) ↔ ((𝜒 ∨ 𝜑) ∧ (𝜒 ∨ 𝜓)))
2		orcom 718	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ (𝜒 ∨ (𝜑 ∧ 𝜓)))
3		orcom 718	. . 3 ⊢ ((𝜑 ∨ 𝜒) ↔ (𝜒 ∨ 𝜑))
4		orcom 718	. . 3 ⊢ ((𝜓 ∨ 𝜒) ↔ (𝜒 ∨ 𝜓))
5	3, 4	anbi12i 456	. 2 ⊢ (((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)) ↔ ((𝜒 ∨ 𝜑) ∧ (𝜒 ∨ 𝜓)))
6	1, 2, 5	3bitr4i 211	1 ⊢ (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ ((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)))

Syntax hints:   ∧ wa 103   ↔ wb 104   ∨ wo 698

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 699

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  orddi  810  pm5.62dc  935  dn1dc  950  suc11g  4534  bj-peano4  13837