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Mirrors > Home > MPE Home > Th. List > ordi | Structured version Visualization version GIF version |
Description: Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Jan-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 28-Nov-2013.) |
Ref | Expression |
---|---|
ordi | ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | jcab 521 | . 2 ⊢ ((¬ 𝜑 → (𝜓 ∧ 𝜒)) ↔ ((¬ 𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) | |
2 | df-or 848 | . 2 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) ↔ (¬ 𝜑 → (𝜓 ∧ 𝜒))) | |
3 | df-or 848 | . . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)) | |
4 | df-or 848 | . . 3 ⊢ ((𝜑 ∨ 𝜒) ↔ (¬ 𝜑 → 𝜒)) | |
5 | 3, 4 | anbi12i 630 | . 2 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)) ↔ ((¬ 𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) |
6 | 1, 2, 5 | 3bitr4i 306 | 1 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 847 |
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 210 df-an 400 df-or 848 |
This theorem is referenced by: ordir 1007 orddi 1010 pm5.63 1020 pm4.43 1023 cadan 1616 undi 4175 undif3 4191 undif4 4367 elnn1uz2 12486 or3di 30483 poxp2 33470 wl-df3-3mintru2 35343 ifpan23 40693 ifpidg 40724 ifpim123g 40733 |
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