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Mirrors > Home > ILE Home > Th. List > 3jaob | GIF version |
Description: Disjunction of 3 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 1166 | . . . 4 ⊢ (𝜑 → (𝜑 ∨ 𝜒 ∨ 𝜃)) | |
2 | 1 | imim1i 60 | . . 3 ⊢ (((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓) → (𝜑 → 𝜓)) |
3 | 3mix2 1167 | . . . 4 ⊢ (𝜒 → (𝜑 ∨ 𝜒 ∨ 𝜃)) | |
4 | 3 | imim1i 60 | . . 3 ⊢ (((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓) → (𝜒 → 𝜓)) |
5 | 3mix3 1168 | . . . 4 ⊢ (𝜃 → (𝜑 ∨ 𝜒 ∨ 𝜃)) | |
6 | 5 | imim1i 60 | . . 3 ⊢ (((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓) → (𝜃 → 𝜓)) |
7 | 2, 4, 6 | 3jca 1177 | . 2 ⊢ (((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓) → ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓))) |
8 | 3jao 1301 | . 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓)) → ((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓)) | |
9 | 7, 8 | impbii 126 | 1 ⊢ (((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∨ w3o 977 ∧ w3a 978 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 |
This theorem is referenced by: (None) |
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