| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > eqeltrrd | Structured version Visualization version GIF version | ||
| Description: Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.) |
| Ref | Expression |
|---|---|
| eqeltrrd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| eqeltrrd.2 | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
| Ref | Expression |
|---|---|
| eqeltrrd | ⊢ (𝜑 → 𝐵 ∈ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqeltrrd.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | 1 | eqcomd 2743 | . 2 ⊢ (𝜑 → 𝐵 = 𝐴) |
| 3 | eqeltrrd.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
| 4 | 2, 3 | eqeltrd 2841 | 1 ⊢ (𝜑 → 𝐵 ∈ 𝐶) |
| Copyright terms: Public domain | W3C validator |