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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 729 | . 2 ⊢ ((𝜒 ∨ (𝜑 ∧ 𝜓)) ↔ ((𝜒 ∨ 𝜑) ∧ (𝜒 ∨ 𝜓))) | |
2 | orcom 647 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ (𝜒 ∨ (𝜑 ∧ 𝜓))) | |
3 | orcom 647 | . . 3 ⊢ ((𝜑 ∨ 𝜒) ↔ (𝜒 ∨ 𝜑)) | |
4 | orcom 647 | . . 3 ⊢ ((𝜓 ∨ 𝜒) ↔ (𝜒 ∨ 𝜓)) | |
5 | 3, 4 | anbi12i 433 | . 2 ⊢ (((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)) ↔ ((𝜒 ∨ 𝜑) ∧ (𝜒 ∨ 𝜓))) |
6 | 1, 2, 5 | 3bitr4i 201 | 1 ⊢ (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ ((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∨ wo 629 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: orddi 733 pm5.62dc 852 dn1dc 867 suc11g 4281 bj-peano4 10080 |
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