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Mirrors > Home > ILE Home > Th. List > ordi | GIF version |
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.) |
Ref | Expression |
---|---|
ordi | ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . . 4 ⊢ ((𝜓 ∧ 𝜒) → 𝜓) | |
2 | 1 | orim2i 751 | . . 3 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) → (𝜑 ∨ 𝜓)) |
3 | simpr 109 | . . . 4 ⊢ ((𝜓 ∧ 𝜒) → 𝜒) | |
4 | 3 | orim2i 751 | . . 3 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) → (𝜑 ∨ 𝜒)) |
5 | 2, 4 | jca 304 | . 2 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) → ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒))) |
6 | orc 702 | . . . 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 701 | . . . 4 ⊢ ((𝜓 ∧ 𝜒) → (𝜑 ∨ (𝜓 ∧ 𝜒))) | |
10 | 8, 9 | jaoian 785 | . . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ 𝜒) → (𝜑 ∨ (𝜓 ∧ 𝜒))) |
11 | 7, 10 | jaodan 787 | . 2 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)) → (𝜑 ∨ (𝜓 ∧ 𝜒))) |
12 | 5, 11 | impbii 125 | 1 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒))) |
Colors of variables: wff set class |
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: ordir 807 orddi 810 pm5.63dc 931 pm4.43 934 orbididc 938 undi 3356 undif4 3457 elnn1uz2 9524 |
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