Theorem ordi 828

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 109	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → 𝜓)
2	1	orim2i 773	. . 3 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) → (𝜑 ∨ 𝜓))
3		simpr 110	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → 𝜒)
4	3	orim2i 773	. . 3 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) → (𝜑 ∨ 𝜒))
5	2, 4	jca 306	. 2 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) → ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))
6		orc 724	. . . 4 ⊢ (𝜑 → (𝜑 ∨ (𝜓 ∧ 𝜒)))
7	6	adantl 277	. . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ 𝜑) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
8	6	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝜒) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
9		olc 723	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
10	8, 9	jaoian 807	. . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ 𝜒) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
11	7, 10	jaodan 809	. 2 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
12	5, 11	impbii 126	1 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))

This proof depends on syntax axioms:   ∧ wa 104   ↔ wb 105   ∨ wo 720

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721

This proof depends on definitions:  df-bi 117

This theorem is used by:  ordir  829  orddi  832  pm5.63dc  959  pm4.43  962  orbididc  966  undi  3479  undif4  3587  elnn1uz2  10017