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Mirrors > Home > MPE Home > Th. List > eleq1i | Structured version Visualization version GIF version |
Description: Inference from equality to equivalence of membership. (Contributed by NM, 21-Jun-1993.) |
Ref | Expression |
---|---|
eleq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
eleq1i | ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | eleq1 2827 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶) |
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