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Mirrors > Home > ILE Home > Th. List > ordir | GIF version |
Description: Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.) |
Ref | Expression |
---|---|
ordir | ⊢ (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ ((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordi 816 | . 2 ⊢ ((𝜒 ∨ (𝜑 ∧ 𝜓)) ↔ ((𝜒 ∨ 𝜑) ∧ (𝜒 ∨ 𝜓))) | |
2 | orcom 728 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ (𝜒 ∨ (𝜑 ∧ 𝜓))) | |
3 | orcom 728 | . . 3 ⊢ ((𝜑 ∨ 𝜒) ↔ (𝜒 ∨ 𝜑)) | |
4 | orcom 728 | . . 3 ⊢ ((𝜓 ∨ 𝜒) ↔ (𝜒 ∨ 𝜓)) | |
5 | 3, 4 | anbi12i 460 | . 2 ⊢ (((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)) ↔ ((𝜒 ∨ 𝜑) ∧ (𝜒 ∨ 𝜓))) |
6 | 1, 2, 5 | 3bitr4i 212 | 1 ⊢ (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ ((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∨ wo 708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: orddi 820 pm5.62dc 945 dn1dc 960 suc11g 4552 bj-peano4 14329 |
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