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| Mirrors > Home > MPE Home > Th. List > eleq2i | Structured version Visualization version GIF version | ||
| Description: Inference from equality to equivalence of membership. (Contributed by NM, 26-May-1993.) |
| Ref | Expression |
|---|---|
| eleq1i.1 | ⊢ 𝐴 = 𝐵 |
| Ref | Expression |
|---|---|
| eleq2i | ⊢ (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eleq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
| 2 | eleq2 2830 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵)) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵) |
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