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| Mirrors > Home > MPE Home > Th. List > eleq2 | Structured version Visualization version GIF version | ||
| Description: Equality implies equivalence of membership. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 20-Nov-2019.) |
| Ref | Expression |
|---|---|
| eleq2 | ⊢ (𝐴 = 𝐵 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | id 22 | . 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵) | |
| 2 | 1 | eleq2d 2827 | 1 ⊢ (𝐴 = 𝐵 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵)) |
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