Theorem ordi 765

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 107	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → 𝜓)
2	1	orim2i 713	. . 3 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) → (𝜑 ∨ 𝜓))
3		simpr 108	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → 𝜒)
4	3	orim2i 713	. . 3 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) → (𝜑 ∨ 𝜒))
5	2, 4	jca 300	. 2 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) → ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))
6		orc 668	. . . 4 ⊢ (𝜑 → (𝜑 ∨ (𝜓 ∧ 𝜒)))
7	6	adantl 271	. . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ 𝜑) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
8	6	adantr 270	. . . 4 ⊢ ((𝜑 ∧ 𝜒) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
9		olc 667	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
10	8, 9	jaoian 744	. . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ 𝜒) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
11	7, 10	jaodan 746	. 2 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
12	5, 11	impbii 124	1 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))

Syntax hints:   ∧ wa 102   ↔ wb 103   ∨ wo 664

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 665

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  ordir  766  orddi  769  pm5.63dc  892  pm4.43  895  orbididc  899  undi  3247  undif4  3345  elnn1uz2  9092