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| Mirrors > Home > MPE Home > Th. List > coeq12d | Structured version Visualization version GIF version | ||
| Description: Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.) |
| Ref | Expression |
|---|---|
| coeq12d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| coeq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
| Ref | Expression |
|---|---|
| coeq12d | ⊢ (𝜑 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | coeq12d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | 1 | coeq1d 5872 | . 2 ⊢ (𝜑 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)) |
| 3 | coeq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
| 4 | 3 | coeq2d 5873 | . 2 ⊢ (𝜑 → (𝐵 ∘ 𝐶) = (𝐵 ∘ 𝐷)) |
| 5 | 2, 4 | eqtrd 2777 | 1 ⊢ (𝜑 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷)) |
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