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| Mirrors > Home > MPE Home > Th. List > eqeltrrid | Structured version Visualization version GIF version | ||
| Description: A membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
| Ref | Expression |
|---|---|
| eqeltrrid.1 | ⊢ 𝐵 = 𝐴 |
| eqeltrrid.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐶) |
| Ref | Expression |
|---|---|
| eqeltrrid | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqeltrrid.1 | . . 3 ⊢ 𝐵 = 𝐴 | |
| 2 | 1 | eqcomi 2746 | . 2 ⊢ 𝐴 = 𝐵 |
| 3 | eqeltrrid.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐶) | |
| 4 | 2, 3 | eqeltrid 2845 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
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