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Mirrors > Home > MPE Home > Th. List > 3jao | Structured version Visualization version GIF version |
Description: Disjunction of three antecedents. (Contributed by NM, 8-Apr-1994.) |
Ref | Expression |
---|---|
3jao | ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓)) → ((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3or 1038 | . 2 ⊢ ((𝜑 ∨ 𝜒 ∨ 𝜃) ↔ ((𝜑 ∨ 𝜒) ∨ 𝜃)) | |
2 | jao 534 | . . . 4 ⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜓) → ((𝜑 ∨ 𝜒) → 𝜓))) | |
3 | jao 534 | . . . 4 ⊢ (((𝜑 ∨ 𝜒) → 𝜓) → ((𝜃 → 𝜓) → (((𝜑 ∨ 𝜒) ∨ 𝜃) → 𝜓))) | |
4 | 2, 3 | syl6 35 | . . 3 ⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜓) → ((𝜃 → 𝜓) → (((𝜑 ∨ 𝜒) ∨ 𝜃) → 𝜓)))) |
5 | 4 | 3imp 1256 | . 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓)) → (((𝜑 ∨ 𝜒) ∨ 𝜃) → 𝜓)) |
6 | 1, 5 | syl5bi 232 | 1 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓)) → ((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 383 ∨ 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: 3jaob 1390 3jaoi 1391 3jaod 1392 |
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