Theorem ordi 805

Description: Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)

Assertion

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

Proof of Theorem ordi

Step	Hyp	Ref	Expression
1		simpl 108	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → 𝜓)
2	1	orim2i 750	. . 3 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) → (𝜑 ∨ 𝜓))
3		simpr 109	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → 𝜒)
4	3	orim2i 750	. . 3 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) → (𝜑 ∨ 𝜒))
5	2, 4	jca 304	. 2 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) → ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))
6		orc 701	. . . 4 ⊢ (𝜑 → (𝜑 ∨ (𝜓 ∧ 𝜒)))
7	6	adantl 275	. . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ 𝜑) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
8	6	adantr 274	. . . 4 ⊢ ((𝜑 ∧ 𝜒) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
9		olc 700	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
10	8, 9	jaoian 784	. . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ 𝜒) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
11	7, 10	jaodan 786	. 2 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
12	5, 11	impbii 125	1 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))

Syntax hints:   ∧ wa 103   ↔ wb 104   ∨ wo 697

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 698

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  ordir  806  orddi  809  pm5.63dc  930  pm4.43  933  orbididc  937  undi  3324  undif4  3425  elnn1uz2  9401