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| Mirrors > Home > MPE Home > Th. List > eqeltrdi | Structured version Visualization version GIF version | ||
| Description: A membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
| Ref | Expression |
|---|---|
| eqeltrdi.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| eqeltrdi.2 | ⊢ 𝐵 ∈ 𝐶 |
| Ref | Expression |
|---|---|
| eqeltrdi | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqeltrdi.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | eqeltrdi.2 | . . 3 ⊢ 𝐵 ∈ 𝐶 | |
| 3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐶) |
| 4 | 1, 3 | eqeltrd 2841 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
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