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| Mirrors > Home > MPE Home > Th. List > 3anbi3d | Structured version Visualization version GIF version | ||
| Description: Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.) |
| Ref | Expression |
|---|---|
| 3anbi1d.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
| Ref | Expression |
|---|---|
| 3anbi3d | ⊢ (𝜑 → ((𝜃 ∧ 𝜏 ∧ 𝜓) ↔ (𝜃 ∧ 𝜏 ∧ 𝜒))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | biidd 262 | . 2 ⊢ (𝜑 → (𝜃 ↔ 𝜃)) | |
| 2 | 3anbi1d.1 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
| 3 | 1, 2 | 3anbi13d 1440 | 1 ⊢ (𝜑 → ((𝜃 ∧ 𝜏 ∧ 𝜓) ↔ (𝜃 ∧ 𝜏 ∧ 𝜒))) |
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