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Mirrors > Home > MPE Home > Th. List > ordir | Structured version Visualization version 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 904 | . 2 ⊢ ((𝜒 ∨ (𝜑 ∧ 𝜓)) ↔ ((𝜒 ∨ 𝜑) ∧ (𝜒 ∨ 𝜓))) | |
2 | orcom 401 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ (𝜒 ∨ (𝜑 ∧ 𝜓))) | |
3 | orcom 401 | . . 3 ⊢ ((𝜑 ∨ 𝜒) ↔ (𝜒 ∨ 𝜑)) | |
4 | orcom 401 | . . 3 ⊢ ((𝜓 ∨ 𝜒) ↔ (𝜒 ∨ 𝜓)) | |
5 | 3, 4 | anbi12i 729 | . 2 ⊢ (((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)) ↔ ((𝜒 ∨ 𝜑) ∧ (𝜒 ∨ 𝜓))) |
6 | 1, 2, 5 | 3bitr4i 291 | 1 ⊢ (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ ((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∨ wo 382 ∧ wa 383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 |
This theorem is referenced by: orddi 909 pm5.62 960 dn1 1000 cadan 1539 elnn0z 11267 ifpim123g 36864 rp-fakeanorass 36877 |
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