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| Mirrors > Home > MPE Home > Th. List > opeq12d | Structured version Visualization version GIF version | ||
| Description: Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
| Ref | Expression |
|---|---|
| opeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| opeq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
| Ref | Expression |
|---|---|
| opeq12d | ⊢ (𝜑 → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐷〉) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | opeq12d.2 | . 2 ⊢ (𝜑 → 𝐶 = 𝐷) | |
| 3 | opeq12 4875 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐷〉) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐷〉) |
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