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| Mirrors > Home > MPE Home > Th. List > 3anbi123d | Structured version Visualization version GIF version | ||
| Description: Deduction joining 3 equivalences to form equivalence of conjunctions. (Contributed by NM, 22-Apr-1994.) |
| Ref | Expression |
|---|---|
| bi3d.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
| bi3d.2 | ⊢ (𝜑 → (𝜃 ↔ 𝜏)) |
| bi3d.3 | ⊢ (𝜑 → (𝜂 ↔ 𝜁)) |
| Ref | Expression |
|---|---|
| 3anbi123d | ⊢ (𝜑 → ((𝜓 ∧ 𝜃 ∧ 𝜂) ↔ (𝜒 ∧ 𝜏 ∧ 𝜁))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bi3d.1 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
| 2 | bi3d.2 | . . . 4 ⊢ (𝜑 → (𝜃 ↔ 𝜏)) | |
| 3 | 1, 2 | anbi12d 632 | . . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) ↔ (𝜒 ∧ 𝜏))) |
| 4 | bi3d.3 | . . 3 ⊢ (𝜑 → (𝜂 ↔ 𝜁)) | |
| 5 | 3, 4 | anbi12d 632 | . 2 ⊢ (𝜑 → (((𝜓 ∧ 𝜃) ∧ 𝜂) ↔ ((𝜒 ∧ 𝜏) ∧ 𝜁))) |
| 6 | df-3an 1089 | . 2 ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) ↔ ((𝜓 ∧ 𝜃) ∧ 𝜂)) | |
| 7 | df-3an 1089 | . 2 ⊢ ((𝜒 ∧ 𝜏 ∧ 𝜁) ↔ ((𝜒 ∧ 𝜏) ∧ 𝜁)) | |
| 8 | 5, 6, 7 | 3bitr4g 314 | 1 ⊢ (𝜑 → ((𝜓 ∧ 𝜃 ∧ 𝜂) ↔ (𝜒 ∧ 𝜏 ∧ 𝜁))) |
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