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Mirrors > Home > MPE Home > Th. List > 3jaob | Structured version Visualization version GIF version |
Description: Disjunction of three antecedents. (Contributed by NM, 13-Sep-2011.) |
Ref | Expression |
---|---|
3jaob | ⊢ (((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3mix1 1230 | . . . 4 ⊢ (𝜑 → (𝜑 ∨ 𝜒 ∨ 𝜃)) | |
2 | 1 | imim1i 63 | . . 3 ⊢ (((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓) → (𝜑 → 𝜓)) |
3 | 3mix2 1231 | . . . 4 ⊢ (𝜒 → (𝜑 ∨ 𝜒 ∨ 𝜃)) | |
4 | 3 | imim1i 63 | . . 3 ⊢ (((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓) → (𝜒 → 𝜓)) |
5 | 3mix3 1232 | . . . 4 ⊢ (𝜃 → (𝜑 ∨ 𝜒 ∨ 𝜃)) | |
6 | 5 | imim1i 63 | . . 3 ⊢ (((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓) → (𝜃 → 𝜓)) |
7 | 2, 4, 6 | 3jca 1242 | . 2 ⊢ (((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓) → ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓))) |
8 | 3jao 1389 | . 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓)) → ((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓)) | |
9 | 7, 8 | impbii 199 | 1 ⊢ (((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∨ w3o 1036 ∧ w3a 1037 |
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 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 |
This theorem is referenced by: (None) |
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