Theorem ordi 729

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 102	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → 𝜓)
2	1	orim2i 678	. . 3 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) → (𝜑 ∨ 𝜓))
3		simpr 103	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → 𝜒)
4	3	orim2i 678	. . 3 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) → (𝜑 ∨ 𝜒))
5	2, 4	jca 290	. 2 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) → ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))
6		orc 633	. . . 4 ⊢ (𝜑 → (𝜑 ∨ (𝜓 ∧ 𝜒)))
7	6	adantl 262	. . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ 𝜑) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
8	6	adantr 261	. . . 4 ⊢ ((𝜑 ∧ 𝜒) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
9		olc 632	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
10	8, 9	jaoian 709	. . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ 𝜒) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
11	7, 10	jaodan 710	. 2 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
12	5, 11	impbii 117	1 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))

Syntax hints:   ∧ wa 97   ↔ wb 98   ∨ wo 629

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  ordir  730  orddi  733  pm5.63dc  853  pm4.43  856  orbididc  860  undi  3185  undif4  3284  elnn1uz2  8544