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Mirrors > Home > MPE Home > Th. List > orddi | Structured version Visualization version GIF version |
Description: Double distributive law for disjunction. (Contributed by NM, 12-Aug-1994.) |
Ref | Expression |
---|---|
orddi | ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ 𝜃)) ↔ (((𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜃)) ∧ ((𝜓 ∨ 𝜒) ∧ (𝜓 ∨ 𝜃)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordir 1005 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∨ (𝜒 ∧ 𝜃)) ∧ (𝜓 ∨ (𝜒 ∧ 𝜃)))) | |
2 | ordi 1004 | . . 3 ⊢ ((𝜑 ∨ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜃))) | |
3 | ordi 1004 | . . 3 ⊢ ((𝜓 ∨ (𝜒 ∧ 𝜃)) ↔ ((𝜓 ∨ 𝜒) ∧ (𝜓 ∨ 𝜃))) | |
4 | 2, 3 | anbi12i 627 | . 2 ⊢ (((𝜑 ∨ (𝜒 ∧ 𝜃)) ∧ (𝜓 ∨ (𝜒 ∧ 𝜃))) ↔ (((𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜃)) ∧ ((𝜓 ∨ 𝜒) ∧ (𝜓 ∨ 𝜃)))) |
5 | 1, 4 | bitri 274 | 1 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ 𝜃)) ↔ (((𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜃)) ∧ ((𝜓 ∨ 𝜒) ∧ (𝜓 ∨ 𝜃)))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∨ wo 845 |
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 206 df-an 397 df-or 846 |
This theorem is referenced by: reuprg 4700 prneimg 4848 wl-cases2-dnf 36185 fzunt 41977 fzuntd 41978 fzunt1d 41979 fzuntgd 41980 |
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