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| Mirrors > Home > MPE Home > Th. List > anbi12d | Structured version Visualization version GIF version | ||
| Description: Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 26-May-1993.) |
| Ref | Expression |
|---|---|
| anbi12d.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
| anbi12d.2 | ⊢ (𝜑 → (𝜃 ↔ 𝜏)) |
| Ref | Expression |
|---|---|
| anbi12d | ⊢ (𝜑 → ((𝜓 ∧ 𝜃) ↔ (𝜒 ∧ 𝜏))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | anbi12d.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
| 2 | 1 | anbi1d 631 | . 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) ↔ (𝜒 ∧ 𝜃))) |
| 3 | anbi12d.2 | . . 3 ⊢ (𝜑 → (𝜃 ↔ 𝜏)) | |
| 4 | 3 | anbi2d 630 | . 2 ⊢ (𝜑 → ((𝜒 ∧ 𝜃) ↔ (𝜒 ∧ 𝜏))) |
| 5 | 2, 4 | bitrd 279 | 1 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) ↔ (𝜒 ∧ 𝜏))) |
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