| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > eqeltrd | Structured version Visualization version GIF version | ||
| Description: Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.) |
| Ref | Expression |
|---|---|
| eqeltrd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| eqeltrd.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐶) |
| Ref | Expression |
|---|---|
| eqeltrd | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqeltrd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐶) | |
| 2 | eqeltrd.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 3 | 2 | eleq1d 2826 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) |
| 4 | 1, 3 | mpbird 257 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
| Copyright terms: Public domain | W3C validator |